Finite Group Databasebeta

← #10691107

Subgroups — #10691107

Order 768.

494 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 105 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#79059C211yesmaximal, characteristic
2#73645C211yesmaximal, characteristic
3#70793C211yesmaximal, characteristic, complemented
4#70757C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9177—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5131—21no—
15#5121—21no—
16#4990—22no—
17#4939—21no—
18#4936C2211yescharacteristic
19C4×C4⋊Dic3—21no—
20#4925C2211yescharacteristic
21#4847—22no—
22#4846C412yescomplemented
23#4842C412yescomplemented
24#4666—21no—
25#4664—21no—
26#4467—22no—
27#4459—22no—
28#4452—21no—
29#4452—21no—
30#3077—31no—
31#2357—31no—
32#1518—31no—
33#1503—31no—
34#1478—31no—
35#1303D311yescharacteristic, complemented
36#1286—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×C4×Dic3—22no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12C2×C412yes—
55C4×D12D412yes—
56#870—21no—
57#869C2×C411yescharacteristic
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8C2×C412yes—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66C3×#95D411yescharacteristic
67C3×#95D411yescharacteristic
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95D412yes—
71#830—21no—
72#821—22no—
73#821—22no—
74#818—22no—
75#817—22no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4×D4—31no—
80#478—62no—
81#476—32no—
82#474—32no—
83#431—31no—
84#431C2×D311yescharacteristic
85#428—61no—
86#423C2×D311yescharacteristic
87#418—61no—
88#390—61no—
89#388—62no—
90#382—31no—
91#381—61no—
92C4×C4⋊C4—61no—
93C4×C4⋊C4C2×D311yescharacteristic
94C4×C22⋊C4—61no—
95#328—62no—
96#328—61no—
97#328—61no—
98D4⋊C8—62no—
99C23×D3—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—22no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—82no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—42no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129#217—41no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—44no—
133(C2×D3)⋊C4—82no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3C22⋊C412yes—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—41no—
145C4×Dic3—82no—
146C2×C3⋊C8—41no—
147C22×D4—31no—
148C23×C4—61no—
149C2×M16—21no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152#125—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×D4—32no—
158C4×D4—32no—
159C4×D4—61no—
160C4×D4—61no—
161#117—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C42—21no—
169C2×C42—62no—
170C2×C42C22×D311yescharacteristic
171#107—62no—
172#106—62no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—61no—
176C4⋊C8C4×D312yes—
177#102—62no—
178#102—62no—
179C22⋊C8—21no—
180C22⋊C8—61no—
181C4×C8—21no—
182C4×C8—61no—
183#95—32no—
184#95—62no—
185#95—61no—
186#95D1211yescharacteristic
187#95D1211yescharacteristic
188C23×C3#12011yescharacteristic
189C22×D3—22no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—41no—
193C22×D3—41no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4#13511yescharacteristic
210C2×C3×C4#13711yescharacteristic
211C2×C3×C4#13711yescharacteristic
212C2×Dic3—22no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219D12—22no—
220D12—22no—
221D12—41no—
222D12—41no—
223C4×D3—82no—
224C4×D3—82no—
225Dic6—81no—
226Dic6—81no—
227C3×C8—42no—
228C3×C8—81no—
229C3⋊C8—81no—
230C24—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×D4—32no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—61no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—242no—
246C22×C4C2×D1211yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249M16—41no—
250M16—41no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—121no—
254C4⋊C4—32no—
255C4⋊C4—42no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—124no—
268C22⋊C4—121no—
269C22⋊C4—242no—
270C42—41no—
271C42—41no—
272C42—121no—
273C42—242no—
274C42C2×C4×D311yescharacteristic
275C42#23511yescharacteristic
276C42D3×D411yescharacteristic
277C42#23911yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#38611yescharacteristic
281C22×C3#45211yescharacteristic
282C22×C3#45211yescharacteristic
283C2×D3—42no—
284C2×D3—42no—
285C2×D3—42no—
286C2×D3—82no—
287C2×D3—82no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—81no—
294C3×C4#41611yescharacteristic
295C3×C4#41911yescharacteristic
296C3×C4#48511yescharacteristic
297C3×C4#48511yescharacteristic
298Dic3—42no—
299Dic3—82no—
300Dic3—81no—
301Dic3—81no—
302C23—62no—
303C23—122no—
304C23—122no—
305C23—121no—
306C23—121no—
307C23#88211yescharacteristic
308Q8—241no—
309Q8—241no—
310D4—62no—
311D4—62no—
312D4—121no—
313D4—121no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—62no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4C4×D1211yescharacteristic
332C2×C4D3×C22⋊C411yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#89311yescharacteristic
335C2×C4#89411yescharacteristic
336C2×C4#89511yescharacteristic
337C2×C4#91211yescharacteristic
338C2×C4#91511yescharacteristic
339C2×C4#91611yescharacteristic
340C2×C4#91811yescharacteristic
341C8—42no—
342C8—81no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#187311yescharacteristic
347C2×C3#190411yescharacteristic
348C2×C3#190611yescharacteristic
349D3—82no—
350C22—21no—
351C22—21no—
352C22—122no—
353C22—122no—
354C22—122no—
355C22—242no—
356C22—242no—
357C22#466511yescharacteristic
358C22#478311yescharacteristic
359C22#480811yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—81no—
366C4—122no—
367C4—242no—
368C4—241no—
369C4—241no—
370C4#477211yescharacteristic
371C4#480211yescharacteristic
372C4#486811yescharacteristic
373C4#488811yescharacteristic
374C3#917711yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—242no—
378C2#6877811yescharacteristic
379C2#6887911yescharacteristic
380C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9177
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #70793C2semidirect
5. #69471C2semidirect
6. C3×#1303C2semidirect
7. #67713C2semidirect
22. #4846C4semidirect
23. #4842C4semidirect
35. #1303D3semidirect
374. C3#9177semidirect

Maximal subgroups

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

subgroupindex
#790592
#736452
#707932
#707572
#694712
C3×#13032
#677132
#91773