Finite Group Databasebeta

← #10697295

Subgroups — #10697295

Order 768.

524 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 109 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#79312C211yesmaximal, characteristic
2#79027C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4#70792C211yesmaximal, characteristic, complemented
5#70459C211yesmaximal, characteristic, complemented
6#69867C211yesmaximal, characteristic
7C3×#1283C211yesmaximal, characteristic, complemented
8#10197—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×#825—22no—
16#5121—22no—
17#5114—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4851—21no—
20#4848—21no—
21#4842—21no—
22#4841—21no—
23#4841—22no—
24#4738—22no—
25#4702—22no—
26#4693—21no—
27#4682—21no—
28#4681—21no—
29#4680—21no—
30#4551—22no—
31#4467—22no—
32#3246—31no—
33#3206—31no—
34#3165—31no—
35#1987—31no—
36#1980—31no—
37#1503—31no—
38#1283D311yescharacteristic, complemented
39C2×#235—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C22×C3×C8D412yes—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C2×#204—24no—
54C2×#204—22no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D412yes—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—21no—
77#825—44no—
78#821—22no—
79#818—24no—
80#817—24no—
81#817—22no—
82#816—24no—
83#816—22no—
84#533—31no—
85#532—31no—
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#498—61no—
90#492—61no—
91#487—61no—
92#476—61no—
93#475—61no—
94#465—62no—
95#461—62no—
96#461—61no—
97#460—61no—
98#448—62no—
99#431C2×D311yescharacteristic
100#428—62no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#104—62no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4—31no—
105#340—62no—
106#328—62no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—41no—
124C2×Dic6—82no—
125C3×M16—22no—
126C3×M16SD1614yescomplemented
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—42no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C4×Dic3—42no—
154#204—84no—
155#204—82no—
156C2×#41—61no—
157C22×Q8—61no—
158C2×SD16—64no—
159C2×SD16—62no—
160C2×M16D1212yes—
161C22×C8D1212yes—
162C4⋊Q8—32no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169#127—31no—
170#125—61no—
171#124—61no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—61no—
175#121—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180#117—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—64no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192#104—124no—
193#103—64no—
194#103—62no—
195#102—62no—
196C22⋊C8D1212yes—
197#97—21no—
198C4×C8—21no—
199C4×C8—21no—
200#95—61no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4C2×SD1612yes—
213C2×C3×C4#13712yes—
214#67—81no—
215#67—81no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—82no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224D12—42no—
225D12—41no—
226D12—41no—
227D12—42no—
228C4×D3—82no—
229Dic6—44no—
230Dic6—41no—
231Dic6—41no—
232Dic6—42no—
233Dic6—82no—
234Dic6—81no—
235C3×C8—24no—
236C3×C8—42no—
237C3×C8—42no—
238#41—122no—
239#41—122no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—121no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4C2×D1211yescharacteristic
254C22×C4C2×D1211yescharacteristic
255C22×C4C2×D1211yescharacteristic
256SD16—244no—
257SD16—242no—
258M16—22no—
259M16#20414yescomplemented
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—42no—
264C4⋊C4—62no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C42—121no—
282C42—122no—
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42#23911yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39211yescharacteristic
290C22×C3#49211yescharacteristic
291C22×C3#49711yescharacteristic
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4#44812yes—
299C3×C4#45511yescharacteristic
300C3×C4#45611yescharacteristic
301C3×C4#46112yes—
302Dic3—82no—
303Dic3—81no—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307C23—121no—
308C23#87411yescharacteristic
309Q8—124no—
310Q8—121no—
311Q8—121no—
312Q8—122no—
313Q8—241no—
314Q8—242no—
315D4—122no—
316D4—121no—
317D4—121no—
318D4—122no—
319D4—241no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4—242no—
333C2×C4#88211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#89511yescharacteristic
337C2×C4#89511yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4C2×#20412yes—
340C2×C4#90912yes—
341C8—24no—
342C8—42no—
343C8—42no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#200711yescharacteristic
347C2×C3#200811yescharacteristic
348C2×C3#201311yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#467111yescharacteristic
356C22#469311yescharacteristic
357C22#471511yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—242no—
362C4—241no—
363C4—241no—
364C4—242no—
365C4—241no—
366C4#473812yes—
367C4#474611yescharacteristic
368C4#475211yescharacteristic
369C4#484112yes—
370C3#1019711yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#6868711yescharacteristic
375C2#6900911yescharacteristic
376C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1283D3
hypocenterC3#10197
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
4. #70792C2semidirect
5. #70459C2semidirect
7. C3×#1283C2semidirect
38. #1283D3semidirect
126. C3×M16SD16semidirect
259. M16#204semidirect
370. C3#10197semidirect

Maximal subgroups

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

subgroupindex
#793122
#790272
#785062
#707922
#704592
#698672
C3×#12832
#101973