Finite Group Databasebeta

← #33122194

Subgroups — #33122194

Order 18688.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989215C211yesmaximal, characteristic
2C2×#17988894C211yesmaximal, characteristic, complemented
3C2×#17988882C211yesmaximal, characteristic, complemented
4#22611269C211yesmaximal, characteristic, complemented
5#22611259C211yesmaximal, characteristic, complemented
6#22611082C211yesmaximal, characteristic, complemented
7#22611072C211yesmaximal, characteristic, complemented
8#22610888C211yesmaximal, characteristic, complemented
9#22610883C211yesmaximal, characteristic, complemented
10#22610244C211yesmaximal, characteristic
11#22610239C211yesmaximal, characteristic
12#22610128C211yesmaximal, characteristic
13#22610118C211yesmaximal, characteristic
14#22609915C211yesmaximal, characteristic
15C73×#2323C211yesmaximal, characteristic
16#17989215—24no—
17#17989215—24no—
18C2×#14556591C2211yescharacteristic
19C2×#14556591C2211yescharacteristic
20C2×#14556590—21no—
21C2×#14556583C2211yescharacteristic
22C2×#14556583C2211yescharacteristic
23C2×#14556582—21no—
24C2×#14556568—21no—
25C2×#14556568—21no—
26C2×#14556550C2211yescharacteristic
27C22×C4⋊Dic73C2211yescharacteristic
28#17988894—24no—
29#17988882—24no—
30C2×#14556524C2211yescharacteristic
31C2×#14556524C2211yescharacteristic
32C2×#14556524C2211yescharacteristic
33C2×#14556522C2211yescharacteristic
34C2×C4⋊#10596831C2211yescharacteristic
35C2×C4⋊#10596831C2211yescharacteristic
36C2×#14556515C2211yescharacteristic
37C2×#14556515C2211yescharacteristic
38C2×#14556515C2211yescharacteristic
39C2×#14556515C2211yescharacteristic
40C2×C22⋊#10596831C2211yescharacteristic
41C2×C22⋊#10596831C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988342C2211yescharacteristic
52C2×C22⋊Q8×C73C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×#117×C73C2211yescharacteristic
55#391×C73C2211yescharacteristic
56#389×C73—21no—
57#389×C73C2211yescharacteristic
58#388×C73—21no—
59#388×C73—21no—
60#387×C73—21no—
61#380×C73C2211yescharacteristic
62C4×C4⋊C4×C73C2211yescharacteristic
63C22×Dic146—21no—
64C22×Dic146—21no—
65C23×Dic73—21no—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70#14556591—22no—
71#14556591—22no—
72#14556591—22no—
73#14556591D414yes—
74#14556590—28no—
75#14556583—22no—
76#14556583—22no—
77#14556583—22no—
78#14556583D414yes—
79#14556582—28no—
80#14556568—28no—
81#14556568—28no—
82#14556550—22no—
83#14556550—22no—
84C2×C4⋊Dic73—22no—
85C2×C4⋊Dic73—22no—
86C2×C4⋊Dic73—21no—
87C2×C4⋊Dic73—21no—
88C2×C4⋊Dic73—21no—
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×C22⋊Dic73—21no—
98C2×C4×Dic73—21no—
99#14556524—22no—
100#14556524—22no—
101#14556524—22no—
102#14556522—22no—
103C4⋊#10596831—22no—
104C4⋊#10596831—22no—
105#14556515—22no—
106#14556515—22no—
107#14556515—22no—
108#14556515—22no—
109C22⋊#10596831—22no—
110C22⋊#10596831—22no—
111C22×Q8×C73C2311yescharacteristic
112C23×C4×C73C2311yescharacteristic
113C22⋊Q8×C73—22no—
114C22⋊Q8×C73Q814yescomplemented
115C4×Q8×C73—22no—
116C4×Q8×C73—22no—
117#117×C73—22no—
118#117×C73—22no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73D411yescharacteristic
123C2×C4⋊C4×C73D411yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C22⋊C4×C73—21no—
128C2×C22⋊C4×C73—21no—
129C2×C22⋊C4×C73D411yescharacteristic
130C2×C22⋊C4×C73D411yescharacteristic
131C2×C42×C73—21no—
132C2×C42×C73—21no—
133C2×C42×C73C2311yescharacteristic
134C2×C42×C73C2311yescharacteristic
135#95×C73—21no—
136#95×C73—21no—
137#95×C73—21no—
138#95×C73—21no—
139#95×C73—21no—
140C2×Dic146—24no—
141C2×Dic146—24no—
142C2×Dic146—81no—
143C2×Dic146—81no—
144C2×Dic146—81no—
145C2×Dic146—81no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—24no—
151C22×Dic73—81no—
152C22×Dic73—81no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C4⋊Dic73—24no—
166C4⋊Dic73—24no—
167C4⋊Dic73—24no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73C2×D412yes—
173C4⋊Dic73C2×D412yes—
174C4⋊Dic73C2×D412yes—
175C4⋊Dic73C2×D412yes—
176#11924007—24no—
177#11924007—24no—
178#11924007—24no—
179#11924007—24no—
180C22⋊Dic73—24no—
181C4×Dic73—24no—
182C24×C73C2×D411yescharacteristic
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73C2×Q812yes—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×Q812yes—
203C22×C4×C73#4112yes—
204C22×C4×C73C2411yescharacteristic
205C4⋊C4×C73—22no—
206C4⋊C4×C73—22no—
207C4⋊C4×C73—24no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—42no—
214C4⋊C4×C73—81no—
215C4⋊C4×C73C2×Q812yes—
216C22⋊C4×C73—24no—
217C22⋊C4×C73—22no—
218C22⋊C4×C73—22no—
219C22⋊C4×C73—81no—
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—42no—
223C42×C73—81no—
224C42×C73C2×D412yes—
225C42×C73C2×D412yes—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73C22⋊Q812yes—
233C23×C73C22×D411yescharacteristic
234Q8×C73—22no—
235Q8×C73—22no—
236Q8×C73—41no—
237Q8×C73—41no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73#12012yes—
267C2×C4×C73#12012yes—
268C2×C4×C73C22⋊Q812yes—
269C2×C4×C73C22×D411yescharacteristic
270C2×C4×C73C22×D411yescharacteristic
271C2×C4×C73C22×Q811yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C732−1+411yescharacteristic
274C2×C4×C732−1+411yescharacteristic
275C2×Dic73—22no—
276C2×Dic73—22no—
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—84no—
288Dic146—84no—
289Dic146—84no—
290#10596831—41no—
291#10596831—41no—
292#10596831—41no—
293#10596831—41no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73C2×#12011yescharacteristic
302C22×C73C2×C22⋊Q811yescharacteristic
303C22×C73#53611yescharacteristic
304C22×C73#57611yescharacteristic
305C22×C73#57711yescharacteristic
306C22×C73#57811yescharacteristic
307C22×C73#57811yescharacteristic
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—42no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—81no—
316C4×C73—81no—
317C4×C73#54811yescharacteristic
318C4×C73#54811yescharacteristic
319C4×C73D4×Q811yescharacteristic
320C4×C73D4×Q811yescharacteristic
321Dic73—42no—
322Dic73—42no—
323Dic73—42no—
324Dic73—42no—
325#24311—731nomaximal, Sylow
326C2×C73—42no—
327C2×C73#240911yescharacteristic
328C2×C73#301111yescharacteristic
329C2×C73#301211yescharacteristic
330C2×C73#308211yescharacteristic
331C2×C73#308311yescharacteristic
332C2×C73#324211yescharacteristic
333C2×C73#324311yescharacteristic
334C2×#544—731no—
335C2×#479—731no—
336C2×#477—731no—
337#2323D7311yescharacteristic
338#2059—731no—
339#2057—731no—
340#2026—731no—
341#2024—731no—
342#1989—731no—
343#1988—731no—
344#1873—731no—
345#1872—731no—
346#1854—731no—
347#1852—731no—
348#1798—731no—
349C73#2431111yescharacteristic, complemented, Sylow
350#544—1464no—
351#544—1464no—
352C2×C4⋊Q8—731no—
353C2×C4⋊Q8—731no—
354C2×C4⋊Q8—1461no—
355C2×#125—731no—
356C2×#125—731no—
357C2×#125—1461no—
358C2×C22⋊Q8—1461no—
359C2×C22⋊Q8—1461no—
360C2×C22⋊Q8C2×D7311yescharacteristic
361C2×C4×Q8C2×D7311yescharacteristic
362C2×#117—731no—
363C2×#117C2×D7311yescharacteristic
364C22×C4⋊C4—731no—
365#479—1464no—
366#477—1464no—
367C2×#107—731no—
368C2×#107—731no—
369C2×#107—731no—
370C2×#106—731no—
371C2×C4⋊C8—731no—
372C2×C4⋊C8—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×C22⋊C8—731no—
378C2×C22⋊C8—731no—
379#391C2×D7311yescharacteristic
380#389—21no—
381#389C2×D7311yescharacteristic
382#388—21no—
383#388—21no—
384#387—21no—
385#380C2×D7311yescharacteristic
386C4×C4⋊C4C2×D7311yescharacteristic
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—1461no—
398C22×Q8—1461no—
399C22×Q8C22×D7311yescharacteristic
400C23×C4—1461no—
401C23×C4C22×D7311yescharacteristic
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C4⋊Q8—734no—
407C4⋊Q8—1462no—
408C4⋊Q8—1462no—
409C4⋊Q8—1468no—
410C4⋊Q8—1462no—
411#125—734no—
412#125—1462no—
413#125—1462no—
414#125—1468no—
415#125—1462no—
416C22⋊Q8—22no—
417C22⋊Q8—1468no—
418C22⋊Q8—1468no—
419C22⋊Q8Dic14614yescomplemented
420C4×Q8—22no—
421C4×Q8—22no—
422#117—22no—
423#117—22no—
424#117—1462no—
425#117—1462no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4—1462no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4C22⋊D7311yescharacteristic
443C2×C4⋊C4C22⋊D7311yescharacteristic
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C4⋊C4C22×D7311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1461no—
450C2×C22⋊C4C22⋊D7311yescharacteristic
451C2×C22⋊C4C22⋊D7311yescharacteristic
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—1461no—
455C2×C42C22×D7311yescharacteristic
456C2×C42C22×D7311yescharacteristic
457#107—1462no—
458#107—1462no—
459#107—1462no—
460#106—1462no—
461C4⋊C8—1462no—
462C4⋊C8—1462no—
463#103—1462no—
464#103—1462no—
465#103—1462no—
466#103—1462no—
467C22⋊C8—1462no—
468C22⋊C8—1462no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474C24C2×C22⋊D7311yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—5841no—
481C2×Q8—5841no—
482C2×Q8—5841no—
483C2×Q8—5841no—
484C2×Q8—1464no—
485C2×Q8—1464no—
486C2×Q8C2×Dic14612yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—5841no—
496C22×C4—5841no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1464no—
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×Dic14612yes—
508C22×C4#1192402812yes—
509C22×C4C23×D7311yescharacteristic
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—732no—
533C4⋊C4—732no—
534C4⋊C4—732no—
535C4⋊C4—732no—
536C4⋊C4—1464no—
537C4⋊C4—1464no—
538C4⋊C4—1464no—
539C4⋊C4—1464no—
540C4⋊C4—1464no—
541C4⋊C4—1464no—
542C4⋊C4—1464no—
543C4⋊C4—1461no—
544C4⋊C4—1461no—
545C4⋊C4—1461no—
546C4⋊C4—1461no—
547C4⋊C4C2×Dic14612yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—81no—
552C22⋊C4—1464no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C42—1464no—
558C42D4×D7312yes—
559C42D4×D7312yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1455657012yes—
567C23C22×C22⋊D7311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—5844no—
573Q8—5844no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—5844no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4#1455655912yes—
616C2×C4#1455655912yes—
617C2×C4#1455657012yes—
618C2×C4C2×D4×D7311yescharacteristic
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4C22×Dic14611yescharacteristic
621C2×C4C2×#1192402811yescharacteristic
622C2×C4#1455663811yescharacteristic
623C2×C4#1455663811yescharacteristic
624C8—2921no—
625C8—2921no—
626C8—2921no—
627C8—2921no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1455655911yescharacteristic
636C22C2×#1455657011yescharacteristic
637C22#1798915811yescharacteristic
638C22#1798939011yescharacteristic
639C22#1798939511yescharacteristic
640C22#1798940211yescharacteristic
641C22#1798940211yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—2922no—
652C4—2922no—
653C4—2922no—
654C4—2922no—
655C4#1798924611yescharacteristic
656C4#1798924611yescharacteristic
657C4D4×Dic14611yescharacteristic
658C4D4×Dic14611yescharacteristic
659C2—42no—
660C2#2261329611yescharacteristic
661C2#2261922411yescharacteristic
662C2#2261923211yescharacteristic
663C2#2261977311yescharacteristic
664C2#2261978211yescharacteristic
665C2#2262119311yescharacteristic
666C2#2262120811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2323C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2323D73
hypocenterC73#24311
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17988894C2semidirect
3. C2×#17988882C2semidirect
4. #22611269C2semidirect
5. #22611259C2semidirect
6. #22611082C2semidirect
7. #22611072C2semidirect
8. #22610888C2semidirect
9. #22610883C2semidirect
114. C22⋊Q8×C73Q8semidirect
349. C73#24311semidirect
419. C22⋊Q8Dic146semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179892152
C2×#179888942
C2×#179888822
#226112692
#226112592
#226110822
#226110722
#226108882
#226108832
#226102442
#226102392
#226101282
#226101182
#226099152
C73×#23232
#2431173