Finite Group Databasebeta

← #10696987

Subgroups — #10696987

Order 768.

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

normal subgroups only

This group has 620 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#79305C211yesmaximal, characteristic
2#78506C211yesmaximal, characteristic, complemented
3C2×#4693C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic, complemented
7C3×#1278C211yesmaximal, characteristic, complemented
8#10153—31nomaximal, Sylow
9C22×#204—22no—
10C2×#874C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#823—22no—
17#5121—22no—
18#4924C2211yescharacteristic
19#4848—22no—
20#4731—22no—
21#4702—22no—
22#4694—21no—
23#4693—22no—
24#4693—21no—
25#4693—22no—
26#4682—21no—
27#4680—21no—
28#4482—22no—
29#4456—22no—
30#3208—31no—
31#3165—31no—
32C2×#492—31no—
33#1965—31no—
34#1549—31no—
35#1493—31no—
36#1278D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—21no—
39C22×D12D412yes—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C2×#204—28no—
51C2×#204—24no—
52C2×#204—48no—
53#874—21no—
54#874—21no—
55#874—21no—
56#874—21no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8D412yes—
73#823—44no—
74#821—24no—
75#818—24no—
76#817—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C22×SD16—62no—
80#533—31no—
81C2×C4⋊Q8—31no—
82C2×#127—31no—
83#501—62no—
84#500—61no—
85#498—61no—
86#492—62no—
87#492—61no—
88#492—62no—
89#486—61no—
90#460—62no—
91#450—62no—
92#431C2×D311yescharacteristic
93#428—62no—
94#423—31no—
95C2×#100—62no—
96C4×M16C2×D311yescharacteristic
97C2×C4×C8C2×D311yescharacteristic
98#332—62no—
99Q8⋊C8—62no—
100C23×D3—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—41no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—44no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—42no—
113C2×D12—42no—
114C2×D12—82no—
115C2×Dic6—21no—
116C2×Dic6—22no—
117C2×Dic6—21no—
118C2×Dic6—44no—
119C2×Dic6—41no—
120C3×M16—22no—
121C3×M16SD1614yescomplemented
122C2×C3×C8—24no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#217—41no—
131C4⋊Dic3—21no—
132C4⋊Dic3—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—22no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141C4×Dic3—41no—
142#208—42no—
143C2×C3⋊C8—41no—
144#204—416no—
145#204—84no—
146C22×Q8—61no—
147C22×D4—32no—
148C22×D4—61no—
149C22×D4—61no—
150C2×SD16—68no—
151C2×SD16—64no—
152C2×SD16—128no—
153C2×M16—61no—
154C2×M16D1212yes—
155C22×C8—21no—
156C4⋊Q8—32no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#127—61no—
163#127—61no—
164#127—61no—
165#127—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176#106—64no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8D1212yes—
180#102—64no—
181#100—124no—
182C22⋊C8—21no—
183#97—21no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8D1212yes—
188C23×C3#12111yescharacteristic
189C22×D3—42no—
190C22×D3—44no—
191C22×D3—82no—
192C22×D3—82no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12711yescharacteristic
201C2×C3×C4C2×SD1612yes—
202C2×C3×C4#13712yes—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—82no—
207D12—44no—
208D12—44no—
209D12—82no—
210D12—82no—
211D12—82no—
212D12—81no—
213D12—81no—
214Dic6—44no—
215Dic6—42no—
216Dic6—42no—
217Dic6—82no—
218C3×C8—24no—
219C3×C8—24no—
220C3×C8—42no—
221C3⋊C8—81no—
222C24—62no—
223C2×Q8—61no—
224C2×Q8—62no—
225C2×Q8—61no—
226C2×Q8—124no—
227C2×Q8—121no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—124no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—121no—
234C2×D4—121no—
235C2×D4—122no—
236C2×D4—122no—
237C2×D4—242no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4#23911yescharacteristic
242SD16—1216no—
243SD16—244no—
244M16—22no—
245M16—122no—
246M16#20414yescomplemented
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—121no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—121no—
263C42—121no—
264C42C2×D1211yescharacteristic
265C42C2×D1211yescharacteristic
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#46112yes—
272C2×D3—82no—
273C2×D3—82no—
274C2×D3—82no—
275C2×D3—164no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4#44812yes—
279C3×C4#45312yes—
280C3×C4#47112yes—
281C3×C4#49211yescharacteristic
282C3×C4#49711yescharacteristic
283Dic3—82no—
284Dic3—81no—
285Dic3—81no—
286C23—122no—
287C23—124no—
288C23—242no—
289C23—242no—
290C23#89511yescharacteristic
291Q8—124no—
292Q8—122no—
293Q8—122no—
294Q8—242no—
295D4—124no—
296D4—124no—
297D4—242no—
298D4—242no—
299D4—241no—
300D4—241no—
301D4—242no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—242no—
308C2×C4#87411yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#89511yescharacteristic
313C2×C4#89511yescharacteristic
314C2×C4C2×#20412yes—
315C2×C4#90912yes—
316C8—24no—
317C8—24no—
318C8—42no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200211yescharacteristic
322C2×C3#201311yescharacteristic
323C2×C3#201911yescharacteristic
324D3—162no—
325C22—21no—
326C22—21no—
327C22—242no—
328C22—242no—
329C22—242no—
330C22—484no—
331C22#467111yescharacteristic
332C22#484112yes—
333C4—21no—
334C4—21no—
335C4—241no—
336C4—242no—
337C4—241no—
338C4#469311yescharacteristic
339C4#471511yescharacteristic
340C4#473812yes—
341C4#482212yes—
342C4#489812yes—
343C3#1015311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—482no—
346C2#6868711yescharacteristic
347C2#6876111yescharacteristic
348C2#6894311yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1278D3
hypocenterC3#10153
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. #78506C2semidirect
4. #70780C2semidirect
6. #69738C2semidirect
7. C3×#1278C2semidirect
36. #1278D3semidirect
121. C3×M16SD16semidirect
246. M16#204semidirect
343. C3#10153semidirect

Maximal subgroups

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

subgroupindex
#793052
#785062
C2×#46932
#707802
#698762
#697382
C3×#12782
#101533