Finite Group Databasebeta

← #10696101

Subgroups — #10696101

Order 768.

518 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 81 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#79344C211yesmaximal, characteristic
2#78507C211yesmaximal, characteristic, complemented
3#74277C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic, complemented
5#69855C211yesmaximal, characteristic
6#69566C211yesmaximal, characteristic, complemented
7C3×#1288C211yesmaximal, characteristic, complemented
8#10023—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#383C2211yescharacteristic
15#5121—21no—
16#5121—21no—
17#4926C2211yescharacteristic
18#4846—21no—
19#4841—21no—
20#4731—21no—
21#4722—21no—
22#4710—22no—
23#4702—21no—
24#4702—21no—
25#4683—22no—
26#4677—21no—
27#4674—21no—
28#4672—21no—
29#4668—22no—
30#4661—21no—
31#4465—22no—
32#4459—22no—
33#4452—22no—
34#3166—31no—
35#3110—31no—
36#2633—31no—
37#1534—31no—
38#1515—31no—
39#1490—31no—
40#1288D311yescharacteristic, complemented
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—22no—
48#926—21no—
49C2×C4⋊Dic3—22no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53#924—21no—
54#924—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×#204—22no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3×C4×C8D412yes—
76C3×#95—21no—
77C3×#95—21no—
78C3×#95D412yes—
79C3⋊#95—22no—
80#830—21no—
81#821—22no—
82#818—22no—
83#818—22no—
84#818—22no—
85#817—22no—
86#817—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#533—31no—
91C2×#125—31no—
92C2×C4×D4—31no—
93#501—61no—
94#501—61no—
95#499—62no—
96#484—61no—
97#474—61no—
98#461—61no—
99#450—61no—
100#440—62no—
101#432—31no—
102#432C2×D311yescharacteristic
103#428—61no—
104#428—61no—
105C4×M16C2×D311yescharacteristic
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383C2×D311yescharacteristic
112#330—62no—
113#328—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—42no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—82no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C3×M16—24no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139(C2×D3)⋊C4—44no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—82no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—41no—
162#204—44no—
163C22×D4—31no—
164C23×C4—61no—
165C2×SD16—62no—
166C2×M16—21no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C42C22×D311yescharacteristic
193#107—62no—
194#107—62no—
195#107—62no—
196#106—62no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200#103—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204#97—21no—
205#97—61no—
206#97D1212yes—
207C4×C8D1212yes—
208#95—21no—
209#95—21no—
210#95—62no—
211#95D1212yes—
212C23×C3#12011yescharacteristic
213C22×D3—22no—
214C22×D3—42no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—42no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239D12—22no—
240D12—22no—
241D12—41no—
242D12—41no—
243C4×D3—82no—
244C4×D3—82no—
245Dic6—42no—
246Dic6—81no—
247C3×C8—24no—
248C3×C8—42no—
249C3⋊C8—81no—
250C24—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—122no—
256C22×C4—22no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—242no—
263C22×C4C2×D1211yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4D3×D411yescharacteristic
266SD16—124no—
267M16—24no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—121no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—242no—
288C22⋊C4—124no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—242no—
293C42—121no—
294C42C2×D1211yescharacteristic
295C42C2×D1211yescharacteristic
296C42D3×D411yescharacteristic
297C42#23911yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#39211yescharacteristic
301C22×C3#45612yes—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—82no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—82no—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317Dic3—81no—
318Dic3—81no—
319C23—62no—
320C23—122no—
321C23—121no—
322C23—121no—
323C23—122no—
324C23#88211yescharacteristic
325Q8—122no—
326Q8—241no—
327D4—62no—
328D4—62no—
329D4—121no—
330D4—121no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—242no—
347C2×C4—242no—
348C2×C4#87411yescharacteristic
349C2×C4#88211yescharacteristic
350C2×C4#88211yescharacteristic
351C2×C4#89511yescharacteristic
352C2×C4#89511yescharacteristic
353C2×C4#89511yescharacteristic
354C8—24no—
355C8—42no—
356C8—241no—
357C2×C3—22no—
358C2×C3#200511yescharacteristic
359C2×C3#200811yescharacteristic
360C2×C3#201011yescharacteristic
361D3—82no—
362C22—21no—
363C22—21no—
364C22—124no—
365C22—122no—
366C22—242no—
367C22—241no—
368C22—241no—
369C22#467111yescharacteristic
370C22#482612yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—241no—
379C4—241no—
380C4—241no—
381C4—241no—
382C4—241no—
383C3#1002311yescharacteristic, complemented, Sylow
384C2—22no—
385C2—242no—
386C2#6870311yescharacteristic
387C2#6876811yescharacteristic
388C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1288D3
hypocenterC3#10023
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
2. #78507C2semidirect
4. #70814C2semidirect
6. #69566C2semidirect
7. C3×#1288C2semidirect
40. #1288D3semidirect
383. C3#10023semidirect

Maximal subgroups

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

subgroupindex
#793442
#785072
#742772
#708142
#698552
#695662
C3×#12882
#100233