Finite Group Databasebeta

← #10691672

Subgroups — #10691672

Order 768.

504 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82626C211yesmaximal, characteristic
2#78605C211yesmaximal, characteristic, complemented
3C3×#1471C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5#69976C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9264—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10#5791C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5218C411yescharacteristic, complemented
14#5217C411yescharacteristic, complemented
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5111—21no—
19C8×#67—21no—
20#5041—21no—
21#5038—21no—
22#5037—21no—
23#5035—21no—
24#4970C411yescharacteristic, complemented
25#4968C411yescharacteristic, complemented
26#4730—21no—
27#4727—21no—
28#67⋊C8—21no—
29#4720—21no—
30C3×#323—21no—
31C3×#323—21no—
32#4537—21no—
33#4537—21no—
34#4470—21no—
35#4467—21no—
36#3188—31no—
37#2977—31no—
38#1504—31no—
39C2×#327—31no—
40#1471D311yescharacteristic, complemented
41#1298—31no—
42#1280—31no—
43C22×C3×D4—21no—
44C22×C3×D4C2×C411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C2×#235D411yescharacteristic
48C2×C3×M16—21no—
49C22×C3×C8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C22⋊C4—21no—
56#953—21no—
57C2×#214—21no—
58C2×#213—21no—
59#930C2×C411yescharacteristic
60#930D411yescharacteristic
61#929—21no—
62C4×#67—21no—
63C4×#67—21no—
64#926D411yescharacteristic
65#924C2×C411yescharacteristic
66#924D411yescharacteristic
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C2×ΓL(1,𝔽25)—21no—
70C2×D3×C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D411yescharacteristic
75C3×C22⋊C8D411yescharacteristic
76#832—21no—
77#832—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830C2×C411yescharacteristic
83#830C2×C411yescharacteristic
84#830D411yescharacteristic
85#830D411yescharacteristic
86(C2×D3)⋊C8—21no—
87(C2×D3)⋊C8—21no—
88#815—21no—
89Dic3⋊C8—21no—
90Dic3⋊C8—21no—
91C8×Dic3—21no—
92#810—21no—
93#809—21no—
94#808—21no—
95#807—21no—
96#525—31no—
97C2×#121C2×D311yescharacteristic
98#484—31no—
99#482—61no—
100#481—31no—
101#466—61no—
102#465—61no—
103#450—31no—
104#449—61no—
105#447—31no—
106#435—61no—
107#435—61no—
108C8×D4—61no—
109C8×D4—61no—
110#433—61no—
111#408—61no—
112#407C2×D311yescharacteristic
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×#95—31no—
117#328—61no—
118#327—61no—
119#327—61no—
120#327—61no—
121#323—21no—
122#323—21no—
123C24×C3—21no—
124C24×C3#4111yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—81no—
130C2×C3×D4—81no—
131C2×C3×D4—81no—
132C22×C3×C4—21no—
133C22×C3×C4—81no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—21no—
141C2×#67—21no—
142C2×#67—21no—
143C22×Dic3—22no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—81no—
147C22×Dic3—81no—
148#235—81no—
149C2×D12C22⋊C411yescharacteristic
150C2×C4×D3—21no—
151C2×Dic6C22⋊C411yescharacteristic
152C3×M16—81no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8—41no—
156C2×C3×C8—81no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161#217—21no—
162#217—21no—
163#217—21no—
164#214—81no—
165#213—81no—
166(C2×D3)⋊C4—21no—
167(C2×D3)⋊C4—21no—
168C4⋊Dic3C22⋊C411yescharacteristic
169C4⋊Dic3C22⋊C411yescharacteristic
170Dic3⋊C4—21no—
171Dic3⋊C4—21no—
172C4×Dic3—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—81no—
178C2×C3⋊C8—81no—
179ΓL(1,𝔽25)—81no—
180D3×C8—81no—
181C2×#41—31no—
182C22×D4—21no—
183C22×D4C4×D311yescharacteristic
184C23×C4—61no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—61no—
187C2×D8—61no—
188C2×M16—21no—
189C2×M16—61no—
190C22×C8—21no—
191C22×C8—61no—
192C22×C8—61no—
193C22×C8—61no—
194#123—61no—
195C22⋊Q8—31no—
196C22⋊Q8—31no—
197#121—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—31no—
202#121—31no—
203#120—61no—
204C4×D4—61no—
205C4×D4—61no—
206#117—31no—
207C2×C4⋊C4C4×D311yescharacteristic
208C2×C22⋊C4—21no—
209#107—61no—
210#106—61no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213#103—61no—
214#102—61no—
215#102—61no—
216#102—61no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—31no—
221C22⋊C8—31no—
222C22⋊C8—31no—
223C22⋊C8—31no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8#6711yescharacteristic
229C22⋊C8#6711yescharacteristic
230#97—61no—
231C4×C8—61no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—81no—
243C23×C3—81no—
244C23×C3C4×D411yescharacteristic
245C23×C3#12111yescharacteristic
246C23×C3#12311yescharacteristic
247C22×D3—21no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—81no—
251C3×D4—82no—
252C3×D4—81no—
253C3×D4—81no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—81no—
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C4×D411yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266#67—41no—
267#67—41no—
268#67—41no—
269#67—41no—
270#67—81no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—42no—
275C2×Dic3—81no—
276C2×Dic3—82no—
277C2×Dic3—81no—
278C2×Dic3—82no—
279D12—41no—
280C4×D3—41no—
281C4×D3—41no—
282Dic6—41no—
283C3×C8—41no—
284C3×C8—41no—
285C3×C8—81no—
286C3⋊C8—41no—
287C3⋊C8—41no—
288C3⋊C8—41no—
289C3⋊C8—41no—
290C24—21no—
291C24#23511yescharacteristic
292#41—241no—
293C2×Q8—31no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—31no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—81no—
303C2×D4—81no—
304C2×D4—81no—
305C22×C4—21no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—81no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—241no—
312C22×C4—241no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4#23711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319SD16—241no—
320D8—241no—
321M16—81no—
322M16—241no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—81no—
331C2×C8—241no—
332C2×C8—241no—
333C2×C8—241no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—31no—
337C4⋊C4—31no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—61no—
343C22⋊C4—61no—
344C22⋊C4—61no—
345C22⋊C4—61no—
346C22⋊C4—61no—
347C42—61no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—161no—
363C22×C3—161no—
364C22×C3#38611yescharacteristic
365C22×C3#44311yescharacteristic
366C22×C3#44411yescharacteristic
367C22×C3#47111yescharacteristic
368C22×C3#47211yescharacteristic
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—81no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380Dic3—82no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—61no—
390C23—81no—
391C23—81no—
392C23#88311yescharacteristic
393C23#88411yescharacteristic
394C23C4×#6711yescharacteristic
395Q8—121no—
396D4—41no—
397D4—41no—
398D4—81no—
399D4—82no—
400D4—81no—
401D4—81no—
402D4—121no—
403D4—121no—
404D4—121no—
405D4—121no—
406D4—121no—
407D4—241no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—81no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—122no—
422C2×C4—241no—
423C2×C4—242no—
424C2×C4—241no—
425C2×C4—242no—
426C2×C4D3×C22⋊C411yescharacteristic
427C2×C4#67⋊C411yescharacteristic
428C2×C4#93711yescharacteristic
429C2×C4#93811yescharacteristic
430C8—41no—
431C8—41no—
432C8—81no—
433C8—121no—
434C8—121no—
435C8—121no—
436C8—121no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—41no—
442C2×C3—161no—
443C2×C3#187611yescharacteristic
444C2×C3#190711yescharacteristic
445C2×C3#191311yescharacteristic
446D3—81no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—161no—
465C22—161no—
466C22#474811yescharacteristic
467C22#474911yescharacteristic
468C22#495611yescharacteristic
469C22#514411yescharacteristic
470C22#514511yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—81no—
476C4—121no—
477C4—121no—
478C4—121no—
479C4—242no—
480C3#926411yescharacteristic, complemented, Sylow
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—161no—
487C2—241no—
488C2#7001111yescharacteristic
489C2#7003411yescharacteristic
490C2#7008811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1471C2
socleC22×C3#386
prime radical#82626C2
cosocleC23×C3×C4C23
hypercenter#1471D3
hypocenterC3#9264
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
2. #78605C2semidirect
3. C3×#1471C2semidirect
4. C2×#4537C2semidirect
5. #69976C2semidirect
6. #67688C2semidirect
7. #67674C2semidirect
13. #5218C4semidirect
14. #5217C4semidirect
24. #4970C4semidirect
25. #4968C4semidirect
40. #1471D3semidirect
480. C3#9264semidirect

Maximal subgroups

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

subgroupindex
#826262
#786052
C3×#14712
C2×#45372
#699762
#676882
#676742
#92643