Finite Group Databasebeta

← #10690393

Subgroups — #10690393

Order 768.

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

normal subgroups only

This group has 436 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#79311C211yesmaximal, characteristic
2#79016C211yesmaximal, characteristic
3#70473C211yesmaximal, characteristic, complemented
4#69858C211yesmaximal, characteristic
5#69857C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#9051—31nomaximal, Sylow
9#5479C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5111—21no—
15C8×#67—21no—
16#4926C2211yescharacteristic
17C2×#803C2211yescharacteristic
18#4845—21no—
19#4840—21no—
20#4839—21no—
21#4836—21no—
22#4829—21no—
23#4828—21no—
24#4739C411yescharacteristic
25#4738C411yescharacteristic
26#4736C411yescharacteristic
27#4721C411yescharacteristic
28C4×Dic12—21no—
29C4×#204—21no—
30#4550—21no—
31#4550—21no—
32#4465—21no—
33#4464—21no—
34#4456—21no—
35#4456—21no—
36#3108—31no—
37#2992—31no—
38#1985—31no—
39#1524—31no—
40#1491—31no—
41#1302—31no—
42#1283D311yescharacteristic, complemented
43C22×Dic6C2×C411yescharacteristic
44C2×C3×M16—21no—
45C22×C3×C8D411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48#930C2×C411yescharacteristic
49#929—21no—
50C4×#67—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52#924C2×C411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C2×Dic12—21no—
57C2×#204—21no—
58C2×ΓL(1,𝔽25)—21no—
59C2×D3×C8—21no—
60#876—21no—
61#875D411yescharacteristic
62C4×D12D411yescharacteristic
63#872—21no—
64#870D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8C2×C411yescharacteristic
74C3×C22⋊C8C2×C411yescharacteristic
75C3×#97—21no—
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78#830—21no—
79#821—21no—
80(C2×D3)⋊C8—21no—
81(C2×D3)⋊C8—21no—
82#818—21no—
83#817—21no—
84#816—21no—
85#816—21no—
86#816—21no—
87#815—21no—
88Dic3⋊C8—21no—
89Dic3⋊C8—21no—
90C8×Dic3—21no—
91C4⋊(C3⋊C8)D411yescharacteristic
92C4⋊(C3⋊C8)D411yescharacteristic
93#803—21no—
94#803—21no—
95#803—21no—
96#529—31no—
97C2×C4×Q8—31no—
98#483—31no—
99#480—31no—
100#479—61no—
101#464—61no—
102#451—31no—
103#448—31no—
104#446—61no—
105C4×Q16—61no—
106C4×SD16—61no—
107#435—61no—
108#435—61no—
109C8×D4—61no—
110#433—61no—
111#432—31no—
112#431C2×D311yescharacteristic
113#424—61no—
114C2×C4⋊C8C2×D311yescharacteristic
115C4×M16C2×D311yescharacteristic
116C2×#97—31no—
117#336—61no—
118#336—61no—
119#330—61no—
120#329—61no—
121Q8⋊C8—61no—
122Q8⋊C8—61no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×D12C22⋊C411yescharacteristic
130C2×C4×D3—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—42no—
134C2×Dic6C22⋊C411yescharacteristic
135C3×M16—22no—
136C3×M16—41no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#217—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163C4×Dic3—21no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171Dic12—42no—
172#204—42no—
173ΓL(1,𝔽25)—81no—
174D3×C8—81no—
175C22×Q8—31no—
176C2×Q16—61no—
177C2×SD16—61no—
178C2×M16—21no—
179C2×M16—61no—
180C22×C8—61no—
181C22×C8—61no—
182C22×C8D1211yescharacteristic
183C22×C8D1211yescharacteristic
184#126—61no—
185#125—31no—
186#124—31no—
187#123—61no—
188C22⋊Q8—31no—
189#121—31no—
190C4×Q8—31no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×D4—31no—
196C4×D4—61no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—61no—
203#106—61no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—31no—
208C4⋊C8—31no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—61no—
212#103—61no—
213#103—61no—
214#102—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8C4×D311yescharacteristic
219C22⋊C8C4×D311yescharacteristic
220#97—21no—
221#97—61no—
222#97—61no—
223#97—61no—
224#97—61no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—61no—
228C23×C3C4×D411yescharacteristic
229C22×D3—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241#67—41no—
242#67—41no—
243C2×Dic3—22no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252D12—41no—
253C4×D3—41no—
254C4×D3—41no—
255Dic6—22no—
256Dic6—22no—
257Dic6—41no—
258Dic6—41no—
259Dic6—41no—
260C3×C8—41no—
261C3×C8—41no—
262C3×C8—42no—
263C3×C8—41no—
264C3×C8—41no—
265C3⋊C8—42no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C2×Q8—31no—
269C2×Q8—61no—
270C2×Q8—61no—
271C2×Q8—122no—
272C2×D4—31no—
273C2×D4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4C2×D1211yescharacteristic
279C22×C4#23511yescharacteristic
280Q16—122no—
281SD16—122no—
282M16—22no—
283M16—41no—
284M16—241no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—62no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—241no—
298C4⋊C4—31no—
299C4⋊C4—31no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C42—61no—
314C42—121no—
315C42—121no—
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42#23911yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3#38611yescharacteristic
323C22×C3#44311yescharacteristic
324C22×C3#44411yescharacteristic
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4#45511yescharacteristic
334C3×C4#45611yescharacteristic
335Dic3—41no—
336Dic3—41no—
337Dic3—42no—
338Dic3—41no—
339Dic3—81no—
340C23—61no—
341C23C4×D1211yescharacteristic
342Q8—62no—
343Q8—62no—
344Q8—121no—
345Q8—121no—
346Q8—121no—
347D4—121no—
348D4—121no—
349D4—121no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—62no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4D3×C22⋊C411yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#89311yescharacteristic
370C2×C4#89411yescharacteristic
371C2×C4#89511yescharacteristic
372C8—41no—
373C8—41no—
374C8—42no—
375C8—41no—
376C8—41no—
377C8—122no—
378C8—121no—
379C8—121no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3#187611yescharacteristic
383C2×C3#188211yescharacteristic
384C2×C3#189111yescharacteristic
385D3—81no—
386C22—21no—
387C22—21no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22#466511yescharacteristic
392C22#470011yescharacteristic
393C22#471711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—121no—
400C4—121no—
401C4—122no—
402C4—121no—
403C4—241no—
404C4#474611yescharacteristic
405C4#475211yescharacteristic
406C3#905111yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—241no—
410C2#6869511yescharacteristic
411C2#6900411yescharacteristic
412C2#6903611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radical#930C2×C4
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9051
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
3. #70473C2semidirect
5. #69857C2semidirect
6. C3×#1283C2semidirect
7. #67742C2semidirect
42. #1283D3semidirect
406. C3#9051semidirect

Maximal subgroups

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

subgroupindex
#793112
#790162
#704732
#698582
#698572
C3×#12832
#677422
#90513