Finite Group Databasebeta

← #10853236

Subgroups — #10853236

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2441C211yesmaximal, characteristic
2#73457C211yesmaximal, characteristic
3#71739C211yesmaximal, characteristic, complemented
4#71738C211yesmaximal, characteristic, complemented
5#71581C211yesmaximal, characteristic
6#71576C211yesmaximal, characteristic
7#69624C211yesmaximal, characteristic, complemented
8#69622C211yesmaximal, characteristic, complemented
9#69602C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69572C211yesmaximal, characteristic, complemented
12#69567C211yesmaximal, characteristic, complemented
13#69513C211yesmaximal, characteristic
14#69501C211yesmaximal, characteristic
15#69450C211yesmaximal, characteristic
16#26018—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C3×#399—21no—
22C3×#396—21no—
23C3×#389—21no—
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#387—21no—
27C3×#384C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×C4×C22⋊C4C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2211yescharacteristic
33C2×#808C2211yescharacteristic
34C2×#807C2211yescharacteristic
35C2×#807C2211yescharacteristic
36#4942—21no—
37Dic3×C22⋊C4—21no—
38C4×C4⋊Dic3C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4654—21no—
45#4654—21no—
46#4529C2211yescharacteristic
47#4529C2211yescharacteristic
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441D311yescharacteristic
61#2345—31no—
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1947—31no—
68#1920—31no—
69#1919—31no—
70#1854—31no—
71#1851—31no—
72#1823—31no—
73#1809—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C3×#125—22no—
78C3×#125—41no—
79C3×#117—22no—
80C3×#117—41no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3Q814yes—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208—41no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127#830—22no—
128#808—22no—
129#808—22no—
130#807—41no—
131#807—41no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—41no—
134C22×M16—31no—
135C2×#125C2×D311yescharacteristic
136C2×#117C2×D311yescharacteristic
137C22×C4⋊C4—31no—
138C2×#107—31no—
139C2×#107—31no—
140C2×#106—31no—
141C2×#106—31no—
142C2×C4⋊C8—31no—
143C2×C4⋊C8—31no—
144C2×C22⋊C8—31no—
145#399—21no—
146#396—21no—
147#389—21no—
148#389—31no—
149#389—31no—
150#389C2×D311yescharacteristic
151#389C2×D311yescharacteristic
152#387—21no—
153#387—61no—
154#387—61no—
155#384C2×D311yescharacteristic
156#380—31no—
157#380—61no—
158#380C2×D311yescharacteristic
159C4×C4⋊C4—31no—
160C4×C4⋊C4—31no—
161C4×C22⋊C4—61no—
162C4×C22⋊C4C2×D311yescharacteristic
163#340—31no—
164#340—31no—
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#340—31no—
175#336—31no—
176#336—31no—
177C24×C3C2×D411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×Q812yes—
192C22×C3×C4#4112yes—
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—24no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—81no—
200C22×Dic3—81no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—81no—
213C3×C4⋊C4—81no—
214C3×C4⋊C4#4112yes—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—81no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—42no—
224#217—42no—
225C4⋊Dic3—24no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3C2×Q812yes—
233C4⋊Dic3C2×Q812yes—
234Dic3⋊C4—81no—
235Dic3⋊C4—81no—
236C4×Dic3—81no—
237C4×Dic3—81no—
238#208—44no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8#4112yes—
251C23×C4—61no—
252C23×C4C22×D311yescharacteristic
253C2×M16—62no—
254C2×M16—62no—
255C2×M16—121no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260#125—22no—
261#125—41no—
262#117—22no—
263#117—41no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4#6711yescharacteristic
287C2×C22⋊C4#6711yescharacteristic
288C2×C42—21no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42C22×D311yescharacteristic
292C2×C42C22×D311yescharacteristic
293#107—121no—
294#107—121no—
295#106—62no—
296#106—62no—
297C4⋊C8—62no—
298C4⋊C8—121no—
299C22⋊C8—62no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95#6711yescharacteristic
309#95#6711yescharacteristic
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3C22⋊Q812yes—
317C23×C3C22×D411yescharacteristic
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4C22⋊Q812yes—
348C2×C3×C4#13712yes—
349C2×C3×C4C22×Q811yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4H(5,𝔽2)11yescharacteristic
354C2×C3×C42−1+411yescharacteristic
355C2×Dic3—22no—
356C2×Dic3—22no—
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—41no—
366C2×Dic3—41no—
367C2×Dic3—84no—
368C3⋊C8—22no—
369C3⋊C8—41no—
370C3⋊C8—41no—
371C3⋊C8—41no—
372C24C2×#6711yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4—64no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—241no—
387C22×C4—241no—
388C22×C4D3×Q812yes—
389C22×C4#23912yes—
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C23×D311yescharacteristic
396M16—124no—
397C2×C8—32no—
398C2×C8—61no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—32no—
417C4⋊C4—32no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—64no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—241no—
432C4⋊C4—241no—
433C4⋊C4#23712yes—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—81no—
438C22⋊C4—122no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—42no—
444C42—241no—
445C42—241no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3C2×C22⋊Q811yescharacteristic
454C22×C3#54611yescharacteristic
455C22×C3#54711yescharacteristic
456C22×C3C2×#13711yescharacteristic
457C22×C3#57511yescharacteristic
458C22×C3#57611yescharacteristic
459C22×C3#57711yescharacteristic
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—41no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#53311yescharacteristic
470C3×C4#54411yescharacteristic
471C3×C4#55111yescharacteristic
472C3×C4#55211yescharacteristic
473Dic3—42no—
474Dic3—42no—
475Dic3—81no—
476Dic3—81no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#94612yes—
484C23C22×#6711yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—244no—
527C2×C4#94612yes—
528C2×C4#95112yes—
529C2×C4C2×#23711yescharacteristic
530C2×C4#100411yescharacteristic
531C2×C4C2×D3×Q811yescharacteristic
532C2×C4C2×#23911yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4#101011yescharacteristic
535C8—62no—
536C8—121no—
537C8—121no—
538C8—121no—
539C2×C3—42no—
540C2×C3#243811yescharacteristic
541C2×C3#307811yescharacteristic
542C2×C3#308111yescharacteristic
543C2×C3#330911yescharacteristic
544C2×C3#333111yescharacteristic
545C2×C3#334611yescharacteristic
546C2×C3#335111yescharacteristic
547C22—22no—
548C22—22no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22#580611yescharacteristic
555C22C2×#94611yescharacteristic
556C22#582211yescharacteristic
557C22C2×#95111yescharacteristic
558C22#582411yescharacteristic
559C22#583511yescharacteristic
560C22#583711yescharacteristic
561C4—22no—
562C4—22no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—122no—
571C4—122no—
572C4—241no—
573C4—241no—
574C4#562211yescharacteristic
575C4#562311yescharacteristic
576C4#563711yescharacteristic
577C4#564211yescharacteristic
578C3#2601811yescharacteristic, complemented, Sylow
579C2—42no—
580C2#8276211yescharacteristic
581C2#8336711yescharacteristic
582C2#8347411yescharacteristic
583C2#8351511yescharacteristic
584C2#8357211yescharacteristic
585C2#8359611yescharacteristic
586C2#8360911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2441C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2441D3
hypocenterC3#26018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
3. #71739C2semidirect
4. #71738C2semidirect
7. #69624C2semidirect
8. #69622C2semidirect
9. #69602C2semidirect
10. #69601C2semidirect
11. #69572C2semidirect
12. #69567C2semidirect
578. C3#26018semidirect

Maximal subgroups

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

subgroupindex
C3×#24412
#734572
#717392
#717382
#715812
#715762
#696242
#696222
#696022
#696012
#695722
#695672
#695132
#695012
#694502
#260183