Finite Group Databasebeta

← #10853383

Subgroups — #10853383

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76303C211yesmaximal, characteristic
2#76061C211yesmaximal, characteristic, complemented
3C3×#2067C211yesmaximal, characteristic
4#70616C211yesmaximal, characteristic, complemented
5#70609C211yesmaximal, characteristic, complemented
6#70459C211yesmaximal, characteristic, complemented
7#70451C211yesmaximal, characteristic, complemented
8#70193C211yesmaximal, characteristic
9#70164C211yesmaximal, characteristic
10#70154C211yesmaximal, characteristic
11Dic3×#102C211yesmaximal, characteristic
12#69638C211yesmaximal, characteristic
13#69601C211yesmaximal, characteristic, complemented
14#68848C211yesmaximal, characteristic, complemented
15#68802C211yesmaximal, characteristic, complemented
16#26030—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C2×#890C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26#5221C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984—21no—
33#4984C2211yescharacteristic
34#4982—21no—
35#4977C2211yescharacteristic
36#4977C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4945—21no—
41#4944—21no—
42#4943—21no—
43Dic3×C22⋊C4—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4656—21no—
47#4646—21no—
48C3×#340C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2550—31no—
61#2357—31no—
62#2067D311yescharacteristic
63#2065—31no—
64#2025—31no—
65#2023—31no—
66#1989—31no—
67#1987—31no—
68#1949—31no—
69#1939—31no—
70#1919—31no—
71#1918—31no—
72#1835—31no—
73#1804—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90D4×Dic3—22no—
91D4×Dic3Q814yescomplemented
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#890—22no—
110#890—41no—
111#869—22no—
112#869—41no—
113C3×#107—22no—
114C3×C4⋊C8—41no—
115C3×#102—22no—
116C3×#102—41no—
117C3×#95—21no—
118#832—22no—
119#832—41no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130Dic3⋊C8—22no—
131Dic3⋊C8—41no—
132C8×Dic3—22no—
133#808—22no—
134C2×C4⋊Q8—31no—
135C2×#125—31no—
136C2×C4×D4—31no—
137C2×#107C2×D311yescharacteristic
138C2×#106—31no—
139C2×C4⋊C8—31no—
140C2×C4⋊C8—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#102—31no—
143C2×#102—31no—
144C2×#102C2×D311yescharacteristic
145C2×#102C2×D311yescharacteristic
146C2×C4×C8—31no—
147#398—61no—
148#395—61no—
149#391—61no—
150#390—31no—
151#390C2×D311yescharacteristic
152#389—31no—
153#389—61no—
154#389C2×D311yescharacteristic
155#388—61no—
156#388—61no—
157#382—31no—
158#382—31no—
159#381—61no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4—31no—
164C4×C22⋊C4—61no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177C24×C3—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C2×Q812yes—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—24no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—81no—
196C22×Dic3—81no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×Dic6—42no—
202C2×Dic6—42no—
203C2×Dic6—81no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4#4112yes—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—81no—
219C3×C42—21no—
220C3×C42—21no—
221#217—24no—
222#217—42no—
223#217—81no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—42no—
231C4⋊Dic3—42no—
232C4⋊Dic3—81no—
233C4⋊Dic3C2×Q812yes—
234C4⋊Dic3#4112yes—
235Dic3⋊C4—22no—
236Dic3⋊C4—41no—
237Dic3⋊C4—42no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—81no—
241C4×Dic3—81no—
242C4×Dic3—81no—
243C4×Dic3C2×Q812yes—
244C4×Dic3#4112yes—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C22×Q8—61no—
252C22×D4C22×D311yescharacteristic
253C23×C4—61no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D311yescharacteristic
258C4⋊Q8—62no—
259C4⋊Q8—121no—
260#125—62no—
261#125—121no—
262C4×D4—34no—
263C4×D4—62no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—61no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#107—22no—
290#106—62no—
291C4⋊C8—41no—
292C4⋊C8—62no—
293C4⋊C8—121no—
294#102—22no—
295#102—41no—
296#102—62no—
297#102—121no—
298C4×C8—62no—
299#95—21no—
300#95—31no—
301#95—31no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310C23×C3—22no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3C22×D411yescharacteristic
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—41no—
321C3×D4—41no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4#13512yes—
337C2×C3×C4C22×Q811yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)11yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—22no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—22no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—84no—
366C2×Dic3C22⋊Q812yes—
367C2×Dic3C22⋊Q812yes—
368Dic6—84no—
369C3×C8—41no—
370C3×C8—41no—
371C3⋊C8—41no—
372C3⋊C8—41no—
373C24—21no—
374C2×Q8—122no—
375C2×Q8—122no—
376C2×Q8—241no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4D3×Q812yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—64no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—241no—
396C22×C4—241no—
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C23×D311yescharacteristic
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—32no—
418C4⋊C4—32no—
419C4⋊C4—42no—
420C4⋊C4—62no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—121no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—241no—
432C4⋊C4#23712yes—
433C22⋊C4—42no—
434C22⋊C4—64no—
435C22⋊C4—81no—
436C22⋊C4—122no—
437C22⋊C4—241no—
438C42—21no—
439C42—21no—
440C42—32no—
441C42—32no—
442C42—61no—
443C42—61no—
444C42—241no—
445C42—241no—
446C42—241no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3C2×C22⋊Q811yescharacteristic
455C22×C3#54611yescharacteristic
456C22×C3#54811yescharacteristic
457C22×C3C2×#13511yescharacteristic
458C22×C3#57511yescharacteristic
459C22×C3#57611yescharacteristic
460C22×C3#57711yescharacteristic
461C3×C4—22no—
462C3×C4—42no—
463C3×C4—41no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4#53311yescharacteristic
467C3×C4#54411yescharacteristic
468C3×C4#55511yescharacteristic
469C3×C4#55611yescharacteristic
470Dic3—22no—
471Dic3—22no—
472Dic3—42no—
473Dic3—41no—
474Dic3—42no—
475Dic3—41no—
476Dic3—81no—
477Dic3—81no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C2×D3×D411yescharacteristic
486Q8—244no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—32no—
498C2×C4—32no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—244no—
530C2×C4#91212yes—
531C2×C4C2×#23711yescharacteristic
532C2×C4#100411yescharacteristic
533C2×C4C2×D3×Q811yescharacteristic
534C2×C4D3×#4111yescharacteristic
535C2×C4D3×#4111yescharacteristic
536C2×C4#101011yescharacteristic
537C8—41no—
538C8—41no—
539C8—121no—
540C8—121no—
541C2×C3—82no—
542C2×C3#243911yescharacteristic
543C2×C3#307611yescharacteristic
544C2×C3#308111yescharacteristic
545C2×C3#330111yescharacteristic
546C2×C3#332211yescharacteristic
547C2×C3#336111yescharacteristic
548C2×C3#337111yescharacteristic
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22#559811yescharacteristic
557C22#561011yescharacteristic
558C22D3×C22⋊Q811yescharacteristic
559C22C2×#91211yescharacteristic
560C22#577011yescharacteristic
561C22#577411yescharacteristic
562C22#577711yescharacteristic
563C4—22no—
564C4—42no—
565C4—41no—
566C4—62no—
567C4—62no—
568C4—81no—
569C4—81no—
570C4—122no—
571C4—121no—
572C4—121no—
573C4—122no—
574C4—241no—
575C4—241no—
576C4#561611yescharacteristic
577C4#562211yescharacteristic
578C4#562311yescharacteristic
579C4#570211yescharacteristic
580C3#2603011yescharacteristic, complemented, Sylow
581C2—82no—
582C2#7728811yescharacteristic
583C2#8031911yescharacteristic
584C2#8058411yescharacteristic
585C2#8073611yescharacteristic
586C2#8084011yescharacteristic
587C2#8096411yescharacteristic
588C2#8116811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2067C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2067D3
hypocenterC3#26030
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. #76061C2semidirect
4. #70616C2semidirect
5. #70609C2semidirect
6. #70459C2semidirect
7. #70451C2semidirect
13. #69601C2semidirect
14. #68848C2semidirect
15. #68802C2semidirect
91. D4×Dic3Q8semidirect
580. C3#26030semidirect

Maximal subgroups

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

subgroupindex
#763032
#760612
C3×#20672
#706162
#706092
#704592
#704512
#701932
#701642
#701542
Dic3×#1022
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#696012
#688482
#688022
#260303