Finite Group Databasebeta

← #10691095

Subgroups — #10691095

Order 768.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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#74228C211yesmaximal, characteristic
3#70792C211yesmaximal, characteristic, complemented
4#70756C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9175—31nomaximal, Sylow
9C2×#893—21no—
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5131—21no—
16#5121—21no—
17#4972—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4847—22no—
21#4846C412yescomplemented
22#4842C412yescomplemented
23#4668—22no—
24#4667—21no—
25D3×#95—21no—
26#4483—22no—
27#4467—21no—
28#4467—21no—
29#4452—22no—
30#3077—31no—
31#2349—31no—
32#1518—31no—
33#1503—31no—
34#1478—31no—
35#1303—31no—
36#1286D311yescharacteristic, complemented
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C4×D3—22no—
40C2×C3×M16—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—22no—
51C2×C4×Dic3—21no—
52#893—42no—
53#893—42no—
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⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8C2×C412yescomplemented
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74#821—22no—
75#821—22no—
76#818—22no—
77#817—22no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×D4—31no—
82C2×C4×D4—61no—
83#478—62no—
84#476—32no—
85#474—32no—
86#431—31no—
87#431C2×D311yescharacteristic
88#428—61no—
89#423C2×D311yescharacteristic
90#418—61no—
91#389—61no—
92#385—62no—
93#382C2×D311yescharacteristic
94#380—61no—
95C4×C4⋊C4—31no—
96C2×#95—61no—
97#328—61no—
98#328—61no—
99#328—62no—
100D4⋊C8—62no—
101C23×D3—21no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×D12SD1612yes—
114C2×D12Q1612yes—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
117C2×C4×D3—44no—
118C2×C4×D3—44no—
119C2×C4×D3—82no—
120C2×C4×D3—82no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133#217—41no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—44no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—42no—
143C4⋊Dic3C22⋊C412yes—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—82no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149C4×Dic3—41no—
150C2×C3⋊C8—41no—
151C22×D4—31no—
152C23×C4—62no—
153C23×C4—61no—
154C2×M16—21no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×D4—32no—
163C4×D4—32no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—122no—
167C4×D4—122no—
168#117—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—62no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—62no—
175C2×C42—61no—
176C2×C42C22×D311yescharacteristic
177#107—62no—
178#106—62no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8C4×D312yescomplemented
183#102—62no—
184#102—62no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C4×C8—21no—
188C4×C8—61no—
189#95—21no—
190#95—31no—
191#95—31no—
192#95—62no—
193#95D1212yes—
194C23×C3#12011yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—42no—
199C22×D3—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4C2×Q1611yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225D12—22no—
226D12—22no—
227D12—22no—
228D12#10314yes—
229C4×D3—82no—
230C4×D3—82no—
231C4×D3—88no—
232Dic6—81no—
233Dic6—81no—
234C3×C8—42no—
235C3×C8—81no—
236C3⋊C8—81no—
237C24—61no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×D4—32no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—22no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4—124no—
253C22×C4—124no—
254C22×C4—242no—
255C22×C4—242no—
256C22×C4C2×D1211yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—121no—
264C4⋊C4—32no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—122no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—242no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—124no—
279C22⋊C4—121no—
280C42—121no—
281C42—121no—
282C42—121no—
283C42C2×C4×D311yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42#23911yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#38611yescharacteristic
290C22×C3#45011yescharacteristic
291C22×C3#45111yescharacteristic
292C2×D3—24no—
293C2×D3—44no—
294C2×D3—44no—
295C2×D3—42no—
296C2×D3—42no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4C2×#10311yescharacteristic
303C3×C4#41711yescharacteristic
304C3×C4#48311yescharacteristic
305C3×C4#48411yescharacteristic
306Dic3—42no—
307Dic3—42no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23—62no—
312C23—62no—
313C23—62no—
314C23—122no—
315C23—122no—
316C23#88211yescharacteristic
317Q8—241no—
318Q8—241no—
319D4—34no—
320D4—62no—
321D4—62no—
322D4—62no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—242no—
339C2×C4—242no—
340C2×C4—248no—
341C2×C4C4×D1211yescharacteristic
342C2×C4D3×C22⋊C411yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#89311yescharacteristic
345C2×C4#89411yescharacteristic
346C2×C4#89511yescharacteristic
347C2×C4#91111yescharacteristic
348C2×C4D3×SD1611yescharacteristic
349C2×C4#91511yescharacteristic
350C2×C4D3×Q1611yescharacteristic
351C8—42no—
352C8—81no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#187211yescharacteristic
357C2×C3#190411yescharacteristic
358C2×C3#190611yescharacteristic
359D3—44no—
360C22—21no—
361C22—21no—
362C22—64no—
363C22—124no—
364C22—124no—
365C22—122no—
366C22—122no—
367C22#466511yescharacteristic
368C22#477611yescharacteristic
369C22#480911yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—122no—
376C4—122no—
377C4—241no—
378C4—241no—
379C4—241no—
380C4#477111yescharacteristic
381C4D3×#10311yescharacteristic
382C4#486211yescharacteristic
383C4#488411yescharacteristic
384C3#917511yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—124no—
388C2#6870411yescharacteristic
389C2#6887811yescharacteristic
390C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9175
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #70792C2semidirect
5. #69546C2semidirect
6. C3×#1286C2semidirect
7. #67715C2semidirect
21. #4846C4semidirect
22. #4842C4semidirect
36. #1286D3semidirect
59. #869C2×C4semidirect
60. #869C2×C4semidirect
65. C3×C4⋊C8C2×C4semidirect
182. C4⋊C8C4×D3semidirect
384. C3#9175semidirect

Maximal subgroups

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

subgroupindex
#790592
#742282
#707922
#707562
#695462
C3×#12862
#677152
#91753