Finite Group Databasebeta

← #33120302

Subgroups — #33120302

Order 18688.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989144C211yesmaximal, characteristic, complemented
2C2×#17988877C211yesmaximal, characteristic, complemented
3C2×#17988792C211yesmaximal, characteristic, complemented
4#22611127C212yesmaximal, complemented
5#22610934C212yesmaximal
6#22610879C211yesmaximal, characteristic, complemented
7#22610854C211yesmaximal, characteristic, complemented
8#22610289C212yesmaximal, complemented
9#22610198C212yesmaximal, complemented
10#22609911C211yesmaximal, characteristic, complemented
11C4×#121×C73C211yesmaximal, characteristic
12C22×#11924016C2212yes—
13#17989144—24no—
14#17989144—24no—
15C2×#14556591C2211yescharacteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556583—21no—
18C2×#14556570—21no—
19C2×#14556570C2212yes—
20C2×C4×Dic146—21no—
21C2×C4×Dic146C2212yes—
22C2×#14556550—21no—
23#17988877—24no—
24#17988792—24no—
25C2×#14556522C2212yes—
26C2×C4⋊#10596831C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556514C2212yes—
29C2×#14556514C2212yes—
30C2×#14556514C2212yes—
31C2×#14556512C2212yes—
32C2×C22⋊#10596831C2212yes—
33#17988349C2212yes—
34#17988349C2212yes—
35#17988342C2212yes—
36C2×#121×C73C2211yescharacteristic
37C2×C4×D4×C73—21no—
38C2×C4×D4×C73C2211yescharacteristic
39C22×C42×C73C2211yescharacteristic
40#390×C73C2211yescharacteristic
41#389×C73C2211yescharacteristic
42#388×C73—21no—
43#386×C73—21no—
44#380×C73C2211yescharacteristic
45C4×C4⋊C4×C73C2211yescharacteristic
46C4×C22⋊C4×C73—21no—
47C22×Dic146—21no—
48C22×Dic146C2312yes—
49C2×#11924016—24no—
50C2×#11924016—24no—
51C2×#11924016—42no—
52C22×#10596831C2312yes—
53C22×#10596831C2312yes—
54#14556591—22no—
55#14556591—22no—
56#14556591D414yescomplemented
57#14556591D414yescomplemented
58#14556583—28no—
59#14556570—28no—
60#14556570—24no—
61#14556570—24no—
62C4×Dic146—28no—
63C4×Dic146—24no—
64C4×Dic146—24no—
65#14556550—28no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2312yes—
70C2×#11924007—22no—
71C2×#11924007—22no—
72C2×C22⋊Dic73—22no—
73C2×C4×Dic73—22no—
74#14556522—24no—
75C4⋊#10596831—22no—
76C4⋊#10596831—22no—
77#14556514—24no—
78#14556514—24no—
79#14556514—42no—
80#14556512—24no—
81C22⋊#10596831—42no—
82C22×D4×C73—21no—
83C22×D4×C73C2311yescharacteristic
84C23×C4×C73—21no—
85C23×C4×C73D411yescharacteristic
86C23×C4×C73D411yescharacteristic
87C23×C4×C73C2311yescharacteristic
88#121×C73—22no—
89#121×C73—22no—
90C4×D4×C73—22no—
91C4×D4×C73—22no—
92C4×D4×C73—44no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73C2311yescharacteristic
95C2×C4⋊C4×C73C2311yescharacteristic
96C2×C4⋊C4×C73C2311yescharacteristic
97C2×C22⋊C4×C73—21no—
98C2×C22⋊C4×C73—21no—
99C2×C22⋊C4×C73—21no—
100C2×C22⋊C4×C73—21no—
101C2×C22⋊C4×C73—21no—
102C2×C42×C73—22no—
103C2×C42×C73—22no—
104C2×C42×C73—21no—
105C2×C42×C73D411yescharacteristic
106C2×C42×C73D411yescharacteristic
107C2×C42×C73C2311yescharacteristic
108C2×C42×C73C2311yescharacteristic
109#95×C73—21no—
110#95×C73—21no—
111#95×C73—21no—
112C2×Dic146—22no—
113C2×Dic146—22no—
114C2×Dic146—22no—
115C2×Dic146—22no—
116C2×Dic146—24no—
117C2×Dic146—22no—
118C2×Dic146—42no—
119C2×Dic146—42no—
120C2×Dic146C2×D414yes—
121C22×Dic73—22no—
122C22×Dic73—22no—
123#11924016—48no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C4⋊Dic73—24no—
131C4⋊Dic73—24no—
132C4⋊Dic73—24no—
133C4⋊Dic73—22no—
134C4⋊Dic73C2×D414yes—
135#11924007—28no—
136#11924007—28no—
137C22⋊Dic73—28no—
138C4×Dic73—28no—
139C24×C73—21no—
140C24×C73C2×D411yescharacteristic
141C2×D4×C73—24no—
142C2×D4×C73—21no—
143C2×D4×C73—21no—
144C2×D4×C73—21no—
145C2×D4×C73—21no—
146C2×D4×C73—21no—
147C2×D4×C73—42no—
148C2×D4×C73—42no—
149C2×D4×C73#4112yes—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—41no—
163C22×C4×C73—41no—
164C22×C4×C73—42no—
165C22×C4×C73—42no—
166C22×C4×C73—42no—
167C22×C4×C73SD1614yescomplemented
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73#4112yes—
181C22⋊C4×C73—24no—
182C22⋊C4×C73—42no—
183C22⋊C4×C73—42no—
184C22⋊C4×C73—42no—
185C22⋊C4×C73—42no—
186C42×C73—22no—
187C42×C73—22no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—42no—
191C42×C73C2×D412yes—
192C42×C73C2×D412yes—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—22no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73C2×SD1612yes—
207C23×C73C22×D411yescharacteristic
208D4×C73—22no—
209D4×C73—22no—
210D4×C73—48no—
211D4×C73—41no—
212D4×C73—41no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—28no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—44no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—84no—
239C2×C4×C73#12012yes—
240C2×C4×C73#12012yes—
241C2×C4×C73C2×SD1612yes—
242C2×C4×C73C2×SD1612yes—
243C2×C4×C73#13712yes—
244C2×C4×C73C22×D411yescharacteristic
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256Dic146—24no—
257Dic146—24no—
258Dic146—48no—
259Dic146—42no—
260Dic146—42no—
261#10596831—42no—
262#10596831—42no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73C2×#12011yescharacteristic
278C22×C73#52511yescharacteristic
279C22×C73#53711yescharacteristic
280C22×C73C22×SD1611yescharacteristic
281C22×C73C2×#13711yescharacteristic
282C22×C73#57511yescharacteristic
283C22×C73#57511yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73#44814yes—
292C4×C73#54611yescharacteristic
293C4×C73#54711yescharacteristic
294C4×C73#54811yescharacteristic
295C4×C73#55011yescharacteristic
296Dic73—44no—
297Dic73—44no—
298#24151—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73—82no—
301C2×C73#240711yescharacteristic
302C2×C73C2×#44811yescharacteristic
303C2×C73#300511yescharacteristic
304C2×C73#310111yescharacteristic
305C2×C73#311611yescharacteristic
306C2×C73#324611yescharacteristic
307C2×C73#325211yescharacteristic
308C2×#533—731no—
309C2×#476—731no—
310C2×#461—731no—
311C4×#121D7311yescharacteristic
312#2035—732no—
313#2000—732no—
314#1987—731no—
315#1981—731no—
316#1880—732no—
317#1865—732no—
318#1797—731no—
319C73#2415111yescharacteristic, complemented, Sylow
320C22×SD16—732no—
321#533—1464no—
322#533—1464no—
323C2×C4⋊Q8—731no—
324C2×C4⋊Q8—731no—
325C2×#125—1461no—
326C2×C22⋊Q8—732no—
327C2×C22⋊Q8—1461no—
328C2×#121C2×D7311yescharacteristic
329C2×C4×Q8—732no—
330C2×C4×Q8—1461no—
331C2×C4×D4—21no—
332C2×C4×D4C2×D7311yescharacteristic
333C2×#117—1461no—
334C22×C42C2×D7311yescharacteristic
335#476—1464no—
336#461—1464no—
337C2×#106—732no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×#103—732no—
341C2×#103—732no—
342C2×#103—732no—
343C2×#102—732no—
344C2×C22⋊C8—732no—
345#390C2×D7311yescharacteristic
346#389C2×D7311yescharacteristic
347#388—21no—
348#386—21no—
349#380C2×D7311yescharacteristic
350C4×C4⋊C4C2×D7311yescharacteristic
351C4×C22⋊C4—21no—
352#340—732no—
353#340—732no—
354#336—732no—
355C22×Q8—732no—
356C22×Q8—1461no—
357C22×D4—21no—
358C22×D4C22×D7311yescharacteristic
359C23×C4—21no—
360C23×C4C22⋊D7311yescharacteristic
361C23×C4C22⋊D7311yescharacteristic
362C23×C4C22×D7311yescharacteristic
363C2×SD16—2922no—
364C2×SD16—1464no—
365C2×SD16—1464no—
366C22×C8—732no—
367C22×C8—732no—
368C4⋊Q8—734no—
369C4⋊Q8—734no—
370C4⋊Q8—1462no—
371C4⋊Q8—1462no—
372#125—1468no—
373C22⋊Q8—1464no—
374C22⋊Q8—1464no—
375C22⋊Q8—1468no—
376#121—22no—
377#121—22no—
378C4×Q8—1464no—
379C4×Q8—1464no—
380C4×Q8—1468no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—44no—
384#117—1468no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—732no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1462no—
390C2×C4⋊C4—1462no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4C22×D7311yescharacteristic
393C2×C4⋊C4C22×D7311yescharacteristic
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1462no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—1462no—
405C2×C42C22⋊D7311yescharacteristic
406C2×C42C22⋊D7311yescharacteristic
407C2×C42C22×D7311yescharacteristic
408C2×C42C22×D7311yescharacteristic
409#106—1464no—
410C4⋊C8—1462no—
411C4⋊C8—1462no—
412#103—2922no—
413#103—1464no—
414#103—1464no—
415#102—1464no—
416C22⋊C8—2922no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—21no—
421C24C2×C22⋊D7311yescharacteristic
422C2×Q8—2922no—
423C2×Q8—2922no—
424C2×Q8—734no—
425C2×Q8—1462no—
426C2×Q8—1462no—
427C2×Q8—1462no—
428C2×Q8—1462no—
429C2×Q8—1464no—
430C2×Q8—1462no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#1192402812yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4#1192401614yescomplemented
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466SD16—2928no—
467C2×C8—1462no—
468C2×C8—1462no—
469C2×C8—1462no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—734no—
480C4⋊C4—1468no—
481C4⋊C4—1468no—
482C4⋊C4—1464no—
483C4⋊C4—1464no—
484C4⋊C4—1464no—
485C4⋊C4—1462no—
486C4⋊C4#1192402812yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—1468no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—1468no—
499C42D4×D7312yes—
500C42D4×D7312yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#1192401612yes—
515C23C22×C22⋊D7311yescharacteristic
516Q8—2928no—
517Q8—2922no—
518Q8—2922no—
519Q8—1464no—
520Q8—1464no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
555C2×C4—2922no—
556C2×C4—1464no—
557C2×C4—1464no—
558C2×C4#1455655912yes—
559C2×C4#1455655912yes—
560C2×C4C2×#1192401612yes—
561C2×C4C2×#1192401612yes—
562C2×C4#1455662112yes—
563C2×C4C2×D4×D7311yescharacteristic
564C2×C4C2×D4×D7311yescharacteristic
565C2×C4C2×#1192402811yescharacteristic
566C2×C4C2×#1192402811yescharacteristic
567C2×C4#1455663711yescharacteristic
568C2×C4#1455663811yescharacteristic
569C8—2922no—
570C8—2922no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1455655911yescharacteristic
586C22#1798909811yescharacteristic
587C22#1798916511yescharacteristic
588C22C22×#1192401611yescharacteristic
589C22C2×#1455662111yescharacteristic
590C22#1798938511yescharacteristic
591C22#1798938511yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2924no—
600C4—2924no—
601C4#1798872514yes—
602C4#1798923311yescharacteristic
603C4#1798924211yescharacteristic
604C4#1798924611yescharacteristic
605C4#1798925811yescharacteristic
606C2—42no—
607C2—82no—
608C2#2261327611yescharacteristic
609C2C2×#1798872511yescharacteristic
610C2#2261917111yescharacteristic
611C2#2261993911yescharacteristic
612C2#2262009311yescharacteristic
613C2#2262124711yescharacteristic
614C2#2262130411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#121×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC4×#121D73
hypocenterC73#24151
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#17989144C2semidirect
2. C2×#17988877C2semidirect
3. C2×#17988792C2semidirect
4. #22611127C2semidirect
6. #22610879C2semidirect
7. #22610854C2semidirect
8. #22610289C2semidirect
9. #22610198C2semidirect
10. #22609911C2semidirect
56. #14556591D4semidirect
57. #14556591D4semidirect
167. C22×C4×C73SD16semidirect
319. C73#24151semidirect
459. C22×C4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179891442
C2×#179888772
C2×#179887922
#226111272
#226109342
#226108792
#226108542
#226102892
#226101982
#226099112
C4×#121×C732
#2415173