Finite Group Databasebeta

← #33114076

Subgroups — #33114076

Order 18688.

1110 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×#17989217C211yesmaximal, characteristic
2C2×#17988626C211yesmaximal, characteristic
3C2×#17988624C211yesmaximal, characteristic
4#22612546C211yesmaximal, characteristic
5#22611272C211yesmaximal, characteristic, complemented
6#22611267C211yesmaximal, characteristic, complemented
7#22611266C211yesmaximal, characteristic, complemented
8#22611264C211yesmaximal, characteristic, complemented
9#22611262C211yesmaximal, characteristic, complemented
10#22611260C211yesmaximal, characteristic, complemented
11#22611258C211yesmaximal, characteristic, complemented
12#22609649C211yesmaximal, characteristic
13#340×D73C211yesmaximal, characteristic
14#22609626C211yesmaximal, characteristic
15C73×#2059C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18#17989217—24no—
19#17989217—24no—
20C2×#14556591C2211yescharacteristic
21C2×#14556590—21no—
22C2×#14556590C2211yescharacteristic
23C2×#14556582C2211yescharacteristic
24C2×#14556581—21no—
25C2×#14556571—21no—
26C2×#14556569—21no—
27C2×#14556549C2211yescharacteristic
28C2×#14556549C2211yescharacteristic
29C2×#14556548C2211yescharacteristic
30C2×C4⋊C4×D73C2211yescharacteristic
31C2×C4⋊C4×D73C2211yescharacteristic
32#17988626—24no—
33#17988624—24no—
34C2×#14556525C2211yescharacteristic
35C2×#14556525C2211yescharacteristic
36C2×#14556524C2211yescharacteristic
37C2×#14556522C2211yescharacteristic
38C2×#14556522C2211yescharacteristic
39C2×#14556522C2211yescharacteristic
40#17988486C2211yescharacteristic
41#17988485C2211yescharacteristic
42#17988484—21no—
43#17988483C2211yescharacteristic
44#17988482—21no—
45#17988446—21no—
46#17988444—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988342C2211yescharacteristic
54#17988341C2211yescharacteristic
55#17988341C2211yescharacteristic
56C2×#125×C73C2211yescharacteristic
57C2×#107×C73C2211yescharacteristic
58C2×#106×C73C2211yescharacteristic
59#389×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62#336×C73C2211yescharacteristic
63C22×Dic146—21no—
64C22×Dic146—21no—
65C22×C4×D73—21no—
66C22×C4×D73—21no—
67C22×C4×D73C2311yescharacteristic
68C2×#11924011—22no—
69C2×#11924011—22no—
70C2×#11924011Q814yescomplemented
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C2×C8×D73—22no—
74C2×C8×D73—22no—
75C2×C8×D73Q814yescomplemented
76#14556591—22no—
77#14556591—22no—
78#14556590—28no—
79#14556590—22no—
80#14556590—22no—
81#14556582—22no—
82#14556582—22no—
83#14556581—28no—
84#14556571—28no—
85#14556569—28no—
86#14556549—22no—
87#14556549—22no—
88#14556549—22no—
89#14556549D414yes—
90#14556548—22no—
91#14556548—22no—
92C2×C4⋊Dic73—21no—
93C2×C4⋊Dic73—21no—
94C2×C4⋊Dic73—21no—
95C2×C4⋊Dic73C2311yescharacteristic
96C2×C4⋊Dic73C2311yescharacteristic
97C2×C4⋊Dic73C2311yescharacteristic
98C2×C4⋊Dic73C2311yescharacteristic
99C2×#11924007—21no—
100C2×#11924007—21no—
101C2×#11924007—21no—
102C2×#11924007—21no—
103C2×#11924007—21no—
104C2×#11924007—21no—
105C2×#11924007D411yescharacteristic
106C2×#11924007D411yescharacteristic
107C4⋊C4×D73—22no—
108C4⋊C4×D73—22no—
109C4⋊C4×D73D414yes—
110C4⋊C4×D73Q814yescomplemented
111C2×#11924005—21no—
112C2×#11924005—21no—
113C2×#11924005D411yescharacteristic
114C2×#11924005D411yescharacteristic
115C2×C4×Dic73—21no—
116C2×C4×Dic73—21no—
117C2×C4×Dic73C2311yescharacteristic
118#14556525—22no—
119#14556525—22no—
120#14556524—22no—
121#14556522—22no—
122#14556522—22no—
123#14556522—22no—
124#14556498—21no—
125#14556498—21no—
126#14556498—21no—
127C22×C8×C73C2311yescharacteristic
128C22×C8×C73C2311yescharacteristic
129#125×C73—22no—
130#125×C73—22no—
131C2×C4⋊C4×C73—21no—
132C2×C4⋊C4×C73—21no—
133C2×C4⋊C4×C73—21no—
134C2×C4⋊C4×C73C2311yescharacteristic
135C2×C4⋊C4×C73C2311yescharacteristic
136C2×C4⋊C4×C73C2311yescharacteristic
137C2×C4⋊C4×C73C2311yescharacteristic
138C2×C42×C73C2311yescharacteristic
139#107×C73—22no—
140#106×C73—22no—
141#95×C73—21no—
142C23×D73C2×D411yescharacteristic
143C2×Dic146—24no—
144C2×Dic146—24no—
145C2×Dic146—81no—
146C2×Dic146—81no—
147C2×Dic146—81no—
148C2×Dic146—81no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156C2×C4×D73—24no—
157C2×C4×D73—24no—
158C2×C4×D73—81no—
159C2×C4×D73—81no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×C4×D73C2×D412yes—
163C2×C4×D73C2×D412yes—
164C2×C4×D73C2×Q812yes—
165C2×C4×D73C2×Q812yes—
166C2×C4×D73C2×Q812yes—
167C2×C4×D73#4112yes—
168#11924011—28no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831C2×Q812yes—
174C2×#10596831C2×Q812yes—
175C8×D73—28no—
176C4⋊Dic73—24no—
177C4⋊Dic73—24no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—21no—
183C4⋊Dic73—81no—
184C4⋊Dic73C2×D412yes—
185C4⋊Dic73C2×D412yes—
186C4⋊Dic73C2×Q812yes—
187#11924007—24no—
188#11924007—24no—
189#11924007—24no—
190#11924007—22no—
191#11924007—22no—
192#11924007—24no—
193#11924007—81no—
194#11924007—81no—
195#11924005—24no—
196#11924005—22no—
197#11924005—22no—
198#11924005—42no—
199C4×Dic73—21no—
200C4×Dic73—21no—
201C4×Dic73—24no—
202C4×Dic73—42no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73C2×D411yescharacteristic
208C22×C4×C73C2×D411yescharacteristic
209C22×C4×C73C2411yescharacteristic
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73C2×Q812yes—
215C2×C8×C73C2×Q812yes—
216C4⋊C4×C73—24no—
217C4⋊C4×C73—24no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73—21no—
223C4⋊C4×C73—81no—
224C4⋊C4×C73C2×D412yes—
225C4⋊C4×C73C2×D412yes—
226C4⋊C4×C73C2×Q812yes—
227C42×C73—21no—
228C42×C73—21no—
229C23×C73C22×D411yescharacteristic
230C22×D73—21no—
231C22×D73—21no—
232C22×D73—21no—
233C22×D73—21no—
234C22×D73#12012yes—
235C22×D73C22⋊Q812yes—
236C22×D73C4⋊Q812yes—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C22×Q811yescharacteristic
258C2×C4×C73C22×Q811yescharacteristic
259C2×C4×C73C22×Q811yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73#12012yes—
280C2×Dic73C22⋊Q812yes—
281C2×Dic73C4⋊Q812yes—
282C4×D73—84no—
283C4×D73—84no—
284C4×D73C22⋊Q814yes—
285C4×D73C22⋊Q814yes—
286C4×D73C4⋊Q814yes—
287C4×D73C4⋊Q814yes—
288Dic146—84no—
289Dic146—84no—
290C8×C73—22no—
291C8×C73—22no—
292#10596831—22no—
293#10596831—22no—
294C22×C73C2×#12011yescharacteristic
295C22×C73C2×C22⋊Q811yescharacteristic
296C22×C73C2×C4⋊Q811yescharacteristic
297C22×C73#57611yescharacteristic
298C22×C73#57711yescharacteristic
299C22×C73#57711yescharacteristic
300C22×C73#57811yescharacteristic
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—22no—
307C2×D73#39514yes—
308C4×C73—42no—
309C4×C73—41no—
310C4×C73—42no—
311C4×C73—41no—
312C4×C73—42no—
313C4×C73—81no—
314C4×C73C2×C22⋊Q811yescharacteristic
315C4×C73C2×C22⋊Q811yescharacteristic
316C4×C73C2×C4⋊Q811yescharacteristic
317C4×C73C2×C4⋊Q811yescharacteristic
318Dic73—42no—
319Dic73—41no—
320Dic73—42no—
321Dic73—42no—
322Dic73—41no—
323Dic73—81no—
324Dic73#39514yes—
325#23557—731nomaximal, Sylow
326C2×C73C2×#39511yescharacteristic
327C2×C73#301211yescharacteristic
328C2×C73#301311yescharacteristic
329C2×C73#308111yescharacteristic
330C2×C73#308211yescharacteristic
331C2×C73#316311yescharacteristic
332C2×C73#316411yescharacteristic
333D73—24no—
334C2×#544—731no—
335C2×#428—731no—
336C2×#427—731no—
337#2316—731no—
338#2059—731no—
339#2059—731no—
340#2059—731no—
341#2059D7311yescharacteristic, complemented
342#2058—731no—
343#2058—731no—
344#2057—731no—
345#2057—731no—
346#1736—731no—
347C2×#340—731no—
348#1729—731no—
349C73#2355711yescharacteristic, complemented, Sylow
350C22×M16—731no—
351C23×C8—731no—
352#544—1464no—
353#544—1464no—
354C2×C4⋊Q8—731no—
355C2×C4⋊Q8—731no—
356C2×C4⋊Q8—1461no—
357C2×#125—731no—
358C2×#125—1461no—
359C2×#125C2×D7311yescharacteristic
360C2×C22⋊Q8—1461no—
361C2×C22⋊Q8—1461no—
362C2×#117—731no—
363C2×#117—731no—
364C2×#117—731no—
365C22×C4⋊C4—731no—
366C22×C4⋊C4—731no—
367#428—1464no—
368#427—1464no—
369C2×#107—731no—
370C2×#107—731no—
371C2×#107—731no—
372C2×#107C2×D7311yescharacteristic
373C2×#106—731no—
374C2×#106—731no—
375C2×#106—731no—
376C2×#106C2×D7311yescharacteristic
377#389—731no—
378#389—731no—
379#389—731no—
380#389—1461no—
381#389—1461no—
382#389C2×D7311yescharacteristic
383#380—1461no—
384#380—1461no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340C2×D7311yescharacteristic
392#340C2×D7311yescharacteristic
393#336—731no—
394#336—731no—
395#336—731no—
396#336C2×D7311yescharacteristic
397C22×Q8—1461no—
398C22×Q8—1461no—
399C23×C4—731no—
400C23×C4—1461no—
401C23×C4—1461no—
402C2×M16—734no—
403C2×M16—1462no—
404C2×M16—1462no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—734no—
408C22×C8—1462no—
409C22×C8—1462no—
410C22×C8C22×D7311yescharacteristic
411C22×C8C22×D7311yescharacteristic
412C4⋊Q8—1462no—
413C4⋊Q8—1462no—
414C4⋊Q8—1462no—
415C4⋊Q8—1468no—
416C4⋊Q8—1462no—
417#125—22no—
418#125—22no—
419#125—1462no—
420#125—1468no—
421#125—1462no—
422C22⋊Q8—1468no—
423C22⋊Q8—1468no—
424#117—734no—
425#117—1462no—
426#117—1462no—
427#117—1462no—
428#117—1462no—
429#117—1462no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—734no—
437C2×C4⋊C4—734no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—731no—
440C2×C4⋊C4—731no—
441C2×C4⋊C4—1462no—
442C2×C4⋊C4—1462no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4—1461no—
446C2×C4⋊C4—1461no—
447C2×C4⋊C4—1461no—
448C2×C4⋊C4—1461no—
449C2×C4⋊C4—1461no—
450C2×C4⋊C4—1461no—
451C2×C4⋊C4—1461no—
452C2×C4⋊C4C22×D7311yescharacteristic
453C2×C4⋊C4C22×D7311yescharacteristic
454C2×C4⋊C4C22×D7311yescharacteristic
455C2×C4⋊C4C22×D7311yescharacteristic
456C2×C22⋊C4—731no—
457C2×C22⋊C4—731no—
458C2×C22⋊C4—1461no—
459C2×C22⋊C4—1461no—
460C2×C42—731no—
461C2×C42—1461no—
462C2×C42—1461no—
463C2×C42C22×D7311yescharacteristic
464#107—22no—
465#107—1462no—
466#107—1462no—
467#107—1462no—
468#106—22no—
469#106—1462no—
470#106—1462no—
471#106—1462no—
472#95—21no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476C24—731no—
477C2×Q8—5841no—
478C2×Q8—5841no—
479C2×Q8—5841no—
480C2×Q8—5841no—
481C2×Q8—1464no—
482C2×Q8—1464no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—5841no—
488C22×C4—5841no—
489C22×C4—5841no—
490C22×C4—5841no—
491C22×C4—731no—
492C22×C4—732no—
493C22×C4—732no—
494C22×C4—732no—
495C22×C4—732no—
496C22×C4—732no—
497C22×C4—732no—
498C22×C4—731no—
499C22×C4—731no—
500C22×C4—1464no—
501C22×C4—1464no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4D4×D7311yescharacteristic
507C22×C4D4×D7311yescharacteristic
508C22×C4C23×D7311yescharacteristic
509M16—1468no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—1468no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8Q8×D7312yes—
522C2×C8Q8×D7312yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—81no—
531C4⋊C4—5841no—
532C4⋊C4—5841no—
533C4⋊C4—5841no—
534C4⋊C4—732no—
535C4⋊C4—732no—
536C4⋊C4—732no—
537C4⋊C4—1464no—
538C4⋊C4—1464no—
539C4⋊C4—1462no—
540C4⋊C4—1462no—
541C4⋊C4—1464no—
542C4⋊C4—1464no—
543C4⋊C4—1464no—
544C4⋊C4—1464no—
545C4⋊C4—1461no—
546C4⋊C4—1461no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4D4×D7312yes—
551C4⋊C4D4×D7312yes—
552C4⋊C4Q8×D7312yes—
553C22⋊C4—2922no—
554C22⋊C4—1462no—
555C22⋊C4—1462no—
556C22⋊C4—1464no—
557C42—21no—
558C42—21no—
559C42—2922no—
560C42—1464no—
561C42—1461no—
562C42—1461no—
563C23—732no—
564C23—732no—
565C23—732no—
566C23—1461no—
567C23—1461no—
568C23—1461no—
569C23—1461no—
570C23C2×D4×D7311yescharacteristic
571Q8—5844no—
572Q8—5844no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—5844no—
601C2×C4—5844no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—734no—
606C2×C4—734no—
607C2×C4—734no—
608C2×C4—734no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1462no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4C2×D4×D7311yescharacteristic
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4C2×Q8×D7311yescharacteristic
621C2×C4C2×Q8×D7311yescharacteristic
622C2×C4C2×Q8×D7311yescharacteristic
623C2×C4#41×D7311yescharacteristic
624C8—22no—
625C8—22no—
626C8—1462no—
627C8—1462no—
628C22—734no—
629C22—1462no—
630C22—1462no—
631C22—1462no—
632C22—1462no—
633C22—1462no—
634C22—1462no—
635C22#120×D7311yescharacteristic
636C22C22⋊Q8×D7311yescharacteristic
637C22C4⋊Q8×D7311yescharacteristic
638C22#1798938911yescharacteristic
639C22#1798939611yescharacteristic
640C22#1798939711yescharacteristic
641C22#1798940111yescharacteristic
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—2922no—
649C4—2922no—
650C4—2921no—
651C4—2921no—
652C4—2922no—
653C4—5841no—
654C4—734no—
655C4C22⋊Q8×D7311yescharacteristic
656C4C22⋊Q8×D7311yescharacteristic
657C4C4⋊Q8×D7311yescharacteristic
658C4C4⋊Q8×D7311yescharacteristic
659C2—1464no—
660C2#395×D7311yescharacteristic
661C2#2261923911yescharacteristic
662C2#2261924411yescharacteristic
663C2#2261975911yescharacteristic
664C2#2261977211yescharacteristic
665C2#2262045811yescharacteristic
666C2#2262046811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2059C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
hypercenter#2059D73
hypocenterC73#23557
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. #22611272C2semidirect
6. #22611267C2semidirect
7. #22611266C2semidirect
8. #22611264C2semidirect
9. #22611262C2semidirect
10. #22611260C2semidirect
11. #22611258C2semidirect
15. C73×#2059C2semidirect
70. C2×#11924011Q8semidirect
75. C2×C8×D73Q8semidirect
110. C4⋊C4×D73Q8semidirect
341. #2059D73semidirect
349. C73#23557semidirect

Maximal subgroups

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

subgroupindex
C2×#179892172
C2×#179886262
C2×#179886242
#226125462
#226112722
#226112672
#226112662
#226112642
#226112622
#226112602
#226112582
#226096492
#340×D732
#226096262
C73×#20592
#2355773