Finite Group Databasebeta

← #10692083

Subgroups — #10692083

Order 768.

612 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 97 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#73412C211yesmaximal, characteristic
3#69868C211yesmaximal, characteristic, complemented
4#69855C211yesmaximal, characteristic
5#69433C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#9323—31nomaximal, Sylow
9C2×C4×D12—21no—
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#376C2211yescharacteristic
15C4×C4⋊Dic3—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4926C2211yescharacteristic
18#4846C412yescomplemented
19#4841C412yescomplemented
20#4731—21no—
21#4722—21no—
22#4710—22no—
23#4666—22no—
24#4666—21no—
25D3×#95—21no—
26#4555—22no—
27#4459—22no—
28#4452—21no—
29#4452—21no—
30#3110—31no—
31#2275—31no—
32#1986—31no—
33#1506—31no—
34#1490—31no—
35#1300D311yescharacteristic, complemented
36#1285—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—22no—
39C22×C4×D3—21no—
40C2×C3×M16—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—21no—
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3D412yes—
52C2×#204—22no—
53C4×D12—28no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12C2×C412yes—
57C4×D12D412yes—
58#870—21no—
59#869C2×C411yescharacteristic, complemented
60#869C2×C411yescharacteristic, complemented
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8C2×C412yescomplemented
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3⋊#95—22no—
72C3⋊#95—21no—
73#830—21no—
74#821—22no—
75#818—22no—
76#816—22no—
77C4⋊(C3⋊C8)—21no—
78#803—21no—
79#533—31no—
80C2×C4×D4—31no—
81C2×C4×D4—61no—
82#484—61no—
83#474—32no—
84#461—32no—
85#450—61no—
86#440—62no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16C2×D311yescharacteristic
90C4×C4⋊C4—31no—
91C4×C4⋊C4—61no—
92C4×C22⋊C4—62no—
93C4×C22⋊C4—61no—
94#376C2×D311yescharacteristic
95C2×#95—61no—
96#339—62no—
97#328—61no—
98#328—61no—
99D4⋊C8—62no—
100C23×D3—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—22no—
108C2×D12—21no—
109C2×D12—24no—
110C2×D12C22⋊C412yes—
111C2×C4×D3—28no—
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—82no—
115C2×C4×D3—82no—
116C2×C4×D3—82no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×M16—22no—
120C3×M16—41no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C42—24no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128#217—41no—
129(C2×D3)⋊C4—28no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132C4⋊Dic3—24no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3C22⋊C412yes—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141C4×Dic3—41no—
142C4×Dic3—42no—
143C4×Dic3—82no—
144C2×C3⋊C8—41no—
145#204—44no—
146C22×D4—31no—
147C23×C4—62no—
148C23×C4—61no—
149C2×SD16—62no—
150C2×M16—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#125—61no—
154C22⋊Q8—61no—
155C22⋊Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158C4×D4—32no—
159C4×D4—32no—
160C4×D4—68no—
161C4×D4—61no—
162C4×D4—61no—
163#117—61no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C42—21no—
169C2×C42—32no—
170C2×C42—62no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—61no—
174C4⋊C8C4×D312yescomplemented
175#103—62no—
176#102—62no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179#97—21no—
180#97—21no—
181#97—61no—
182C4×C8—21no—
183#95—21no—
184#95—62no—
185#95—61no—
186#95D1211yescharacteristic
187#95D1211yescharacteristic
188C23×C3#12011yescharacteristic
189C22×D3—24no—
190C22×D3—22no—
191C22×D3—42no—
192C22×D3—41no—
193C22×D3—41no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×Dic3—24no—
211C2×Dic3—22no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217D12—24no—
218D12—22no—
219D12#10414yes—
220C4×D3—82no—
221C4×D3—82no—
222C4×D3—88no—
223Dic6—42no—
224Dic6—81no—
225C3×C8—42no—
226C3×C8—42no—
227C3⋊C8—81no—
228C24—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×D4—32no—
232C2×D4—64no—
233C2×D4—61no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—68no—
237C22×C4—62no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—242no—
242C22×C4—242no—
243C22×C4—242no—
244C22×C4C2×D1211yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247SD16—124no—
248M16—22no—
249M16—41no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—121no—
253C4⋊C4—32no—
254C4⋊C4—64no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—68no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—121no—
266C42—24no—
267C42—122no—
268C42—121no—
269C42—242no—
270C42C2×C4×D311yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23911yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#38611yescharacteristic
277C22×C3#45511yescharacteristic
278C22×C3#45611yescharacteristic
279C2×D3—24no—
280C2×D3—44no—
281C2×D3—44no—
282C2×D3—42no—
283C2×D3—42no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4—42no—
291C3×C4C2×#10412yes—
292Dic3—44no—
293Dic3—42no—
294Dic3—81no—
295Dic3—81no—
296C23—64no—
297C23—62no—
298C23—122no—
299C23—121no—
300C23—121no—
301C23#88211yescharacteristic
302Q8—122no—
303Q8—241no—
304D4—34no—
305D4—64no—
306D4—62no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—64no—
318C2×C4—62no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—242no—
325C2×C4—242no—
326C2×C4—248no—
327C2×C4C4×D1211yescharacteristic
328C2×C4D3×C22⋊C411yescharacteristic
329C2×C4#88411yescharacteristic
330C2×C4#89311yescharacteristic
331C2×C4#89411yescharacteristic
332C2×C4#89511yescharacteristic
333C8—42no—
334C8—42no—
335C8—241no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3#188211yescharacteristic
339C2×C3#189611yescharacteristic
340C2×C3#190411yescharacteristic
341D3—44no—
342C22—21no—
343C22—21no—
344C22—64no—
345C22—124no—
346C22—124no—
347C22—122no—
348C22—122no—
349C22#466511yescharacteristic
350C22#482511yescharacteristic
351C22#482611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—42no—
359C4—124no—
360C4—122no—
361C4—241no—
362C4—241no—
363C4D3×#10412yes—
364C3#932311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—124no—
368C2#6875611yescharacteristic
369C2#6877511yescharacteristic
370C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9323
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #69868C2semidirect
5. #69433C2semidirect
6. C3×#1300C2semidirect
7. #67750C2semidirect
18. #4846C4semidirect
19. #4841C4semidirect
35. #1300D3semidirect
59. #869C2×C4semidirect
60. #869C2×C4semidirect
63. C3×C4⋊C8C2×C4semidirect
174. C4⋊C8C4×D3semidirect
364. C3#9323semidirect

Maximal subgroups

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

subgroupindex
#793442
#734122
#698682
#698552
#694332
C3×#13002
#677502
#93233