Finite Group Databasebeta

← #10692403

Subgroups — #10692403

Order 768.

390 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 85 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#79324C211yesmaximal, characteristic
2#78510C211yesmaximal, characteristic
3#69856C211yesmaximal, characteristic, complemented
4#69566C211yesmaximal, characteristic, complemented
5#69565C211yesmaximal, characteristic
6C3×#1288C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#9369—31nomaximal, Sylow
9#5479C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5128—21no—
15#5127—21no—
16#4926C2211yescharacteristic
17#4925C2211yescharacteristic
18#4847—21no—
19#4829—21no—
20#4737—21no—
21#4723—21no—
22#4714C411yescharacteristic, complemented
23#4714C411yescharacteristic, complemented
24#4706—21no—
25#4702C411yescharacteristic, complemented
26#4702C411yescharacteristic, complemented
27#4701—21no—
28#4690—21no—
29#4689—21no—
30#4456—21no—
31#4456—21no—
32#4452—21no—
33#4452—21no—
34#4452—21no—
35#4451—21no—
36#3240—31no—
37#2967—31no—
38#1521—31no—
39#1515—31no—
40#1485—31no—
41#1288D311yescharacteristic, complemented
42#1287—31no—
43C2×C3×M16—21no—
44C2×C3×C42C2311yescharacteristic
45#930D411yescharacteristic
46#929—21no—
47C4×#67—21no—
48#926—21no—
49#924—21no—
50#924—21no—
51#924D411yescharacteristic
52C2×ΓL(1,𝔽25)—21no—
53C2×D3×C8—21no—
54#876—21no—
55#875C2×C411yescharacteristic
56C4×D12D411yescharacteristic
57#872—21no—
58#870—21no—
59#870C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61#869C2×C411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#97C2×C411yescharacteristic
69C3×#97C2×C411yescharacteristic
70C3×C4×C8D411yescharacteristic
71C3×C4×C8D411yescharacteristic
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75#830—21no—
76#830D411yescharacteristic
77#830D411yescharacteristic
78#821—21no—
79#821—21no—
80(C2×D3)⋊C8—21no—
81(C2×D3)⋊C8—21no—
82#818—21no—
83#818—21no—
84#817—21no—
85#817—21no—
86#816—21no—
87#816—21no—
88#815—21no—
89Dic3⋊C8—21no—
90Dic3⋊C8—21no—
91C8×Dic3—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96#533—31no—
97#529—31no—
98#501—31no—
99#501—31no—
100#489—31no—
101#489—31no—
102#485—61no—
103#482—61no—
104#479—61no—
105#478—61no—
106#446—61no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#432—31no—
112#432C2×D311yescharacteristic
113#431—31no—
114#405—61no—
115C4×M16C2×D311yescharacteristic
116#383C2×D311yescharacteristic
117#328—61no—
118#328—61no—
119#328—61no—
120#327—61no—
121Q8⋊C8—61no—
122Q8⋊C8—61no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—21no—
128C2×D12C22⋊C411yescharacteristic
129C2×C4×D3—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6C22⋊C411yescharacteristic
133C3×M16—42no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#217—21no—
143#217—41no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3C22⋊C411yescharacteristic
153C4⋊Dic3C22⋊C411yescharacteristic
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—21no—
161C4×Dic3—41no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—41no—
165ΓL(1,𝔽25)—42no—
166D3×C8—42no—
167C2×M16—21no—
168C2×M16—61no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#126—61no—
173#125—31no—
174#125—61no—
175#124—31no—
176#123—61no—
177C22⋊Q8—31no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180#121—31no—
181C4×Q8—31no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—31no—
185C4×D4—61no—
186#117—61no—
187#117—61no—
188C2×C42C22×D311yescharacteristic
189#107—61no—
190#107—61no—
191#106—61no—
192#106—61no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198#103—61no—
199#103—61no—
200#102—61no—
201#102—61no—
202C22⋊C8—21no—
203C22⋊C8—31no—
204C22⋊C8—31no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#97—21no—
209#97—61no—
210#97—61no—
211#97C4×D311yescharacteristic
212#97C4×D311yescharacteristic
213C4×C8—61no—
214C4×C8—61no—
215C4×C8D1211yescharacteristic
216C4×C8D1211yescharacteristic
217#95—21no—
218#95—21no—
219#95—21no—
220C23×C3#12111yescharacteristic
221C22×D3—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234#67—81no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240D12—41no—
241C4×D3—41no—
242C4×D3—41no—
243Dic6—41no—
244Dic6—42no—
245Dic6—81no—
246C3×C8—22no—
247C3×C8—41no—
248C3×C8—41no—
249C3×C8—41no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8—81no—
253C2×Q8—31no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×D4—31no—
257C2×D4—61no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4#23511yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263M16—42no—
264M16—122no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—122no—
272C2×C8—121no—
273C4⋊C4—31no—
274C4⋊C4—31no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—121no—
291C42—61no—
292C42—121no—
293C42C2×C4×D311yescharacteristic
294C42C2×D1211yescharacteristic
295C42D3×D411yescharacteristic
296C42#23911yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#38611yescharacteristic
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—161no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3—81no—
315Dic3—81no—
316C23—61no—
317C23#88411yescharacteristic
318Q8—121no—
319Q8—122no—
320Q8—241no—
321D4—121no—
322D4—241no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4C4×D1211yescharacteristic
337C2×C4D3×C22⋊C411yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#89311yescharacteristic
340C2×C4#89411yescharacteristic
341C2×C4#89511yescharacteristic
342C8—22no—
343C8—41no—
344C8—41no—
345C8—41no—
346C8—121no—
347C8—121no—
348C8—241no—
349C2×C3—41no—
350C2×C3#188111yescharacteristic
351C2×C3#190611yescharacteristic
352C2×C3#190911yescharacteristic
353D3—81no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22#466511yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4—161no—
371C4—241no—
372C4—241no—
373C3#936911yescharacteristic, complemented, Sylow
374C2—41no—
375C2—241no—
376C2#6871211yescharacteristic
377C2#6877811yescharacteristic
378C2#6903511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9369
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
3. #69856C2semidirect
4. #69566C2semidirect
6. C3×#1288C2semidirect
7. #67756C2semidirect
22. #4714C4semidirect
23. #4714C4semidirect
25. #4702C4semidirect
26. #4702C4semidirect
41. #1288D3semidirect
373. C3#9369semidirect

Maximal subgroups

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

subgroupindex
#793242
#785102
#698562
#695662
#695652
C3×#12882
#677562
#93693