Finite Group Databasebeta

← #10691663

Subgroups — #10691663

Order 768.

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

normal subgroups only

This group has 478 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#78618C211yesmaximal, characteristic
2#78605C211yesmaximal, characteristic
3#69976C211yesmaximal, characteristic
4#69972C211yesmaximal, characteristic, complemented
5C2×#4467C211yesmaximal, characteristic, complemented
6C3×#1298C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9263—31nomaximal, Sylow
9#5791C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3⋊#95C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5116—21no—
17#5114—21no—
18#5111—21no—
19C8×#67—21no—
20#4739C411yescharacteristic, complemented
21#4738C411yescharacteristic, complemented
22#4736C411yescharacteristic, complemented
23#4733—21no—
24#4730—21no—
25#4727—21no—
26#67⋊C8—21no—
27#4723—21no—
28#4721C411yescharacteristic, complemented
29#4720—21no—
30#4470—21no—
31#4469—21no—
32#4469—21no—
33#4467—21no—
34#4467—21no—
35#4467—21no—
36#3190—31no—
37#2977—31no—
38#1504—31no—
39C2×#328—31no—
40#1472—31no—
41#1298D311yescharacteristic, complemented
42#1280—31no—
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C2×#235D411yescharacteristic
46C22×Dic6C2×C411yescharacteristic
47C2×C3×M16—21no—
48C22×C3×C8—21no—
49C22×C3×C8—21no—
50#953—21no—
51C2×#217—21no—
52#930C2×C411yescharacteristic
53#930D411yescharacteristic
54#929—21no—
55C4×#67—21no—
56C4×#67—21no—
57#926D411yescharacteristic
58C2×C4⋊Dic3C2×C411yescharacteristic
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63#924C2×C411yescharacteristic
64#924D411yescharacteristic
65C2×Dic3⋊C4—21no—
66C22×C3⋊C8—21no—
67C2×Dic12—21no—
68C2×#204—21no—
69C2×ΓL(1,𝔽25)—21no—
70C2×D3×C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8C2×C411yescharacteristic
76C3×C22⋊C8C2×C411yescharacteristic
77C3×C22⋊C8D411yescharacteristic
78C3×C22⋊C8D411yescharacteristic
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830D411yescharacteristic
83#830D411yescharacteristic
84#821—21no—
85(C2×D3)⋊C8—21no—
86(C2×D3)⋊C8—21no—
87#818—21no—
88#817—21no—
89#816—21no—
90#816—21no—
91#816—21no—
92#815—21no—
93Dic3⋊C8—21no—
94Dic3⋊C8—21no—
95C8×Dic3—21no—
96#525—31no—
97C2×C22⋊Q8—31no—
98#485—61no—
99#483—31no—
100#480—31no—
101#467—61no—
102#465—61no—
103#452—61no—
104#451—31no—
105#448—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⋊C8C2×D311yescharacteristic
115C2×C22⋊C8C2×D311yescharacteristic
116C2×#95—31no—
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121#324—61no—
122#324—61no—
123C24×C3#4111yescharacteristic
124C22×C3×C4—81no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132C2×#67—21no—
133C2×#67—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C22×Dic3—81no—
140C22×Dic3—81no—
141#235—81no—
142C2×D12C22⋊C411yescharacteristic
143C2×C4×D3—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—81no—
147C2×Dic6C22⋊C411yescharacteristic
148C3×M16—81no—
149C2×C3×C8—21no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8—41no—
154C2×C3×C8—81no—
155C2×C3×C8—81no—
156#217—21no—
157#217—21no—
158#217—21no—
159#217—41no—
160#217—41no—
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3C22⋊C411yescharacteristic
166C4⋊Dic3C22⋊C411yescharacteristic
167Dic3⋊C4—21no—
168Dic3⋊C4—21no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—81no—
175Dic12—81no—
176#204—81no—
177ΓL(1,𝔽25)—81no—
178D3×C8—81no—
179C2×#41—31no—
180C22×Q8—31no—
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C2×Q16—61no—
184C2×SD16—61no—
185C2×M16—21no—
186C2×M16—61no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—61no—
190C22×C8—61no—
191#123—61no—
192C22⋊Q8—31no—
193C22⋊Q8—31no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198#121—31no—
199#121—31no—
200#120—61no—
201C4×D4—61no—
202C4×D4—61no—
203#117—31no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C22⋊C4—61no—
207#107—61no—
208#106—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—61no—
212#103—61no—
213#103—61no—
214#102—61no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—31no—
220C22⋊C8—31no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8C4×D311yescharacteristic
225C22⋊C8C4×D311yescharacteristic
226C22⋊C8D1211yescharacteristic
227C22⋊C8D1211yescharacteristic
228#97—61no—
229C4×C8—61no—
230#95—61no—
231#95—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—41no—
237C23×C3C4×D411yescharacteristic
238C23×C3#12111yescharacteristic
239C23×C3#12311yescharacteristic
240C22×D3—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251#67—41no—
252#67—41no—
253#67—41no—
254#67—41no—
255#67—81no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—81no—
265C2×Dic3—82no—
266C2×Dic3—81no—
267C2×Dic3—81no—
268C2×Dic3—82no—
269D12—41no—
270C4×D3—41no—
271C4×D3—41no—
272Dic6—41no—
273Dic6—41no—
274Dic6—41no—
275Dic6—81no—
276C3×C8—41no—
277C3×C8—41no—
278C3×C8—41no—
279C3×C8—41no—
280C3×C8—81no—
281C3⋊C8—41no—
282C3⋊C8—41no—
283C24#23911yescharacteristic
284#41—241no—
285C2×Q8—31no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—241no—
289C2×D4—31no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—61no—
297C22×C4—81no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—241no—
301C22×C4—241no—
302C22×C4C2×C4×D311yescharacteristic
303C22×C4C2×D1211yescharacteristic
304C22×C4#23511yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308Q16—241no—
309SD16—241no—
310M16—81no—
311M16—241no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—81no—
320C2×C8—81no—
321C2×C8—241no—
322C2×C8—241no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C42—61no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3#38611yescharacteristic
351C22×C3#44311yescharacteristic
352C22×C3#44411yescharacteristic
353C22×C3#47111yescharacteristic
354C22×C3#47311yescharacteristic
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—41no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362Dic3—41no—
363Dic3—41no—
364Dic3—41no—
365Dic3—82no—
366Dic3—81no—
367Dic3—81no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—61no—
374C23#89311yescharacteristic
375C23#89411yescharacteristic
376C23#89511yescharacteristic
377Q8—121no—
378Q8—121no—
379Q8—121no—
380Q8—241no—
381D4—121no—
382D4—121no—
383D4—121no—
384D4—121no—
385D4—121no—
386D4—241no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—122no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—241no—
404C2×C4—242no—
405C2×C4—241no—
406C2×C4—241no—
407C2×C4—242no—
408C2×C4C4×D1211yescharacteristic
409C2×C4D3×C22⋊C411yescharacteristic
410C2×C4#88211yescharacteristic
411C2×C4#88411yescharacteristic
412C8—41no—
413C8—41no—
414C8—41no—
415C8—41no—
416C8—81no—
417C8—121no—
418C8—121no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3#187611yescharacteristic
425C2×C3#190811yescharacteristic
426C2×C3#191311yescharacteristic
427D3—81no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22#466511yescharacteristic
443C22#489411yescharacteristic
444C22#489511yescharacteristic
445C22#489611yescharacteristic
446C22#489811yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—121no—
452C4—121no—
453C4—121no—
454C4—242no—
455C4—241no—
456C4—241no—
457C3#926311yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—241no—
464C2#6894611yescharacteristic
465C2#6894911yescharacteristic
466C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1298D3
hypocenterC3#9263
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
4. #69972C2semidirect
5. C2×#4467C2semidirect
6. C3×#1298C2semidirect
7. #67674C2semidirect
20. #4739C4semidirect
21. #4738C4semidirect
22. #4736C4semidirect
28. #4721C4semidirect
41. #1298D3semidirect
457. C3#9263semidirect

Maximal subgroups

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

subgroupindex
#786182
#786052
#699762
#699722
C2×#44672
C3×#12982
#676742
#92633