Finite Group Databasebeta

← #10692454

Subgroups — #10692454

Order 768.

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

normal subgroups only

This group has 430 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#79358C211yesmaximal, characteristic
2#79341C211yesmaximal, characteristic
3#70822C211yesmaximal, characteristic, complemented
4#69489C211yesmaximal, characteristic, complemented
5#69470C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#9376—31nomaximal, Sylow
9#5479C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4926C2211yescharacteristic
15C2×C4⋊(C3⋊C8)C2211yescharacteristic
16#4849—21no—
17#4841—21no—
18#4840—21no—
19#4839—21no—
20#4739—21no—
21#4738—21no—
22#4730—21no—
23#67⋊C8—21no—
24#4715C412yescomplemented
25#4706—21no—
26#4703C412yescomplemented
27#4701—21no—
28#4529—21no—
29#4529—21no—
30#4486—21no—
31#4486—21no—
32#4452—21no—
33#4451—21no—
34#3218—31no—
35#2992—31no—
36#1987—31no—
37#1556—31no—
38#1509—31no—
39#1290—31no—
40#1285D311yescharacteristic, complemented
41C22×Dic6—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#930D411yescharacteristic
46#929—21no—
47C4×#67—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
52C22×C3⋊C8D411yescharacteristic
53C2×Dic12—22no—
54C2×#204—22no—
55C2×ΓL(1,𝔽25)—21no—
56C2×D3×C8—21no—
57#876—21no—
58#875C2×C411yescharacteristic
59C4×D12D411yescharacteristic
60#872—21no—
61#870C2×C411yescharacteristic
62#869—21no—
63#869—21no—
64#869—21no—
65#869C2×C412yes—
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3×#97C2×C412yes—
74C3×#95—21no—
75#830—21no—
76(C2×D3)⋊C8—21no—
77(C2×D3)⋊C8—21no—
78#816—22no—
79#816—22no—
80#815—21no—
81Dic3⋊C8—21no—
82Dic3⋊C8—21no—
83C8×Dic3—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88C2×C4⋊Q8—31no—
89#529—31no—
90#497—32no—
91#491—32no—
92#462—61no—
93#461—61no—
94#451—61no—
95#448—61no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432C2×D311yescharacteristic
103#432C2×D311yescharacteristic
104#424—61no—
105C2×C4⋊C8—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107#340—61no—
108#340—61no—
109#332—61no—
110#332—61no—
111#328—61no—
112#327—61no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—21no—
118C22×Dic3—41no—
119C2×D12C22⋊C411yescharacteristic
120C2×C4×D3—21no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C2×Dic6C22⋊C411yescharacteristic
125C2×C3×C8—22no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C42—81no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3C22⋊C411yescharacteristic
145C4⋊Dic3C22⋊C411yescharacteristic
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150C4×Dic3—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156Dic12—82no—
157#204—82no—
158ΓL(1,𝔽25)—81no—
159D3×C8—81no—
160C22×Q8—61no—
161C2×Q16—62no—
162C2×SD16—62no—
163C2×M16—61no—
164C22×C8—31no—
165C22×C8—31no—
166C22×C8—61no—
167C4⋊Q8—32no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171#126—61no—
172#125—31no—
173#124—31no—
174#123—61no—
175C22⋊Q8—31no—
176#121—31no—
177C4×Q8—31no—
178C4×D4—31no—
179C4×D4—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8D1211yescharacteristic
193C4⋊C8D1211yescharacteristic
194#103—62no—
195#103—62no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#97—21no—
202#97—61no—
203#97—61no—
204#97C4×D312yes—
205C4×C8—61no—
206#95—21no—
207C23×C3#12111yescharacteristic
208C22×D3—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223#67—41no—
224#67—41no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230D12—41no—
231C4×D3—41no—
232C4×D3—41no—
233Dic6—41no—
234Dic6—42no—
235Dic6—42no—
236Dic6—82no—
237C3×C8—42no—
238C3×C8—41no—
239C3×C8—41no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—81no—
243C2×Q8—31no—
244C2×Q8—62no—
245C2×Q8—121no—
246C2×Q8—242no—
247C2×D4—31no—
248C2×D4—61no—
249C22×C4—41no—
250C22×C4—61no—
251C22×C4—121no—
252C22×C4#23511yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255Q16—242no—
256SD16—242no—
257M16—241no—
258C2×C8—22no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—241no—
267C4⋊C4—22no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C42—61no—
285C42—81no—
286C42C2×C4×D311yescharacteristic
287C42C2×D1211yescharacteristic
288C42D3×D411yescharacteristic
289C42#23911yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#38611yescharacteristic
293C22×C3#47111yescharacteristic
294C22×C3#47311yescharacteristic
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4#45511yescharacteristic
305C3×C4#45611yescharacteristic
306Dic3—41no—
307Dic3—41no—
308Dic3—41no—
309Dic3—82no—
310C23—61no—
311C23#88411yescharacteristic
312Q8—121no—
313Q8—122no—
314Q8—122no—
315Q8—242no—
316D4—121no—
317D4—121no—
318D4—121no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—122no—
333C2×C4—242no—
334C2×C4C4×D1211yescharacteristic
335C2×C4D3×C22⋊C411yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#89311yescharacteristic
338C2×C4#89411yescharacteristic
339C2×C4#89511yescharacteristic
340C8—42no—
341C8—41no—
342C8—41no—
343C8—121no—
344C8—121no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#188211yescharacteristic
349C2×C3#190811yescharacteristic
350C2×C3#191011yescharacteristic
351D3—81no—
352C22—21no—
353C22—21no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22#466511yescharacteristic
358C22#474811yescharacteristic
359C22#475511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—121no—
367C4—121no—
368C4—121no—
369C4—242no—
370C4#482511yescharacteristic
371C4#482611yescharacteristic
372C3#937611yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—241no—
376C2#6871311yescharacteristic
377C2#6877511yescharacteristic
378C2#6900711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9376
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
3. #70822C2semidirect
4. #69489C2semidirect
6. C3×#1285C2semidirect
7. #67755C2semidirect
24. #4715C4semidirect
26. #4703C4semidirect
40. #1285D3semidirect
372. C3#9376semidirect

Maximal subgroups

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

subgroupindex
#793582
#793412
#708222
#694892
#694702
C3×#12852
#677552
#93763