Finite Group Databasebeta

← #10696983

Subgroups — #10696983

Order 768.

620 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 107 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#83207C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic
3#79305C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5#70521C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#10153—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—22no—
11C2×#874C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#823—22no—
16#5085—22no—
17#5085—22no—
18#5081—22no—
19#5080—21no—
20#5078—21no—
21#4964—22no—
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4848—22no—
25#4731—22no—
26#4694—21no—
27#4693—21no—
28C3×#332—22no—
29#4484—22no—
30#3208—31no—
31#3165—31no—
32C2×#492—31no—
33#1965—31no—
34#1549D311yescharacteristic, complemented
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12D412yes—
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#215—28no—
50C2×#215—48no—
51#926—21no—
52#924—21no—
53#924—21no—
54C2×#208D412yes—
55C22×C3⋊C8—21no—
56C2×#204—24no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71#823—44no—
72#821—24no—
73#808—24no—
74#807—24no—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8D412yes—
79C22×SD16—62no—
80#533—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#127—31no—
83#501—62no—
84#500—61no—
85#498—61no—
86#492—61no—
87#492—62no—
88#492—62no—
89#486—61no—
90#460—62no—
91#450—62no—
92#431C2×D311yescharacteristic
93#428—62no—
94#423C2×D311yescharacteristic
95C2×#100—62no—
96C4×M16—31no—
97C2×C4×C8—31no—
98#332—22no—
99Q8⋊C8—62no—
100C23×D3—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×Q8—41no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—44no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—82no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×M16—42no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#217—41no—
132#215—416no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139C4×Dic3—41no—
140#208—22no—
141#208SD1614yescomplemented
142C2×C3⋊C8—24no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145#204—84no—
146C22×Q8—21no—
147C22×D4—32no—
148C22×D4—61no—
149C22×D4—61no—
150C2×SD16—64no—
151C2×SD16—68no—
152C2×SD16—128no—
153C2×M16—21no—
154C2×M16—32no—
155C22×C8—61no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8D1212yes—
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—21no—
173C2×C4⋊C4—21no—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176#106—64no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8#6712yes—
180#102—64no—
181#100—124no—
182C22⋊C8—21no—
183#97—61no—
184C4×C8—21no—
185C4×C8—32no—
186C4×C8—61no—
187C4×C8—61no—
188C23×C3#12111yescharacteristic
189C22×D3—42no—
190C22×D3—44no—
191C22×D3—82no—
192C22×D3—82no—
193C3×Q8—44no—
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—42no—
198C2×C3×C4—82no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12711yescharacteristic
205C2×C3×C4C2×SD1612yes—
206C2×C3×C4#13712yes—
207C2×Dic3—41no—
208C2×Dic3—41no—
209D12—44no—
210D12—44no—
211D12—82no—
212D12—82no—
213D12—82no—
214D12—81no—
215D12—81no—
216Dic6—42no—
217Dic6—42no—
218C3×C8—42no—
219C3×C8—81no—
220C3⋊C8—24no—
221C3⋊C8—24no—
222C24—62no—
223C2×Q8—22no—
224C2×Q8—44no—
225C2×Q8—41no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—124no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—242no—
238C22×C4—41no—
239C22×C4#23511yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—1216no—
243SD16—244no—
244M16—34no—
245M16—42no—
246M16—62no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—64no—
250C2×C8—62no—
251C2×C8—62no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—121no—
263C42—121no—
264C42C2×D1211yescharacteristic
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42C2×#6711yescharacteristic
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—82no—
279C3×C4#44812yes—
280C3×C4#45312yes—
281C3×C4#47112yes—
282C3×C4#49211yescharacteristic
283C3×C4#49711yescharacteristic
284Dic3—81no—
285Dic3—81no—
286C23—122no—
287C23—124no—
288C23—242no—
289C23—242no—
290C23#88411yescharacteristic
291Q8—44no—
292Q8—82no—
293Q8—122no—
294Q8—122no—
295D4—124no—
296D4—124no—
297D4—242no—
298D4—241no—
299D4—241no—
300D4—242no—
301D4—242no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#88211yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4D3×SD1612yes—
312C2×C4#91812yes—
313C2×C4#93011yescharacteristic
314C2×C4#93711yescharacteristic
315C2×C4#94011yescharacteristic
316C8—42no—
317C8—64no—
318C8—64no—
319C8—81no—
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#481812yes—
332C22#496011yescharacteristic
333C4—21no—
334C4—21no—
335C4—82no—
336C4—241no—
337C4—241no—
338C4#475412yes—
339C4#480612yes—
340C4#517511yescharacteristic
341C4#519011yescharacteristic
342C4#519812yes—
343C3#1015311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—482no—
346C2#7011711yescharacteristic
347C2#7028211yescharacteristic
348C2#7032411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1549D3
hypocenterC3#10153
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #83207C2semidirect
4. C3×#1549C2semidirect
6. #70501C2semidirect
7. #67739C2semidirect
34. #1549D3semidirect
141. #208SD16semidirect
343. C3#10153semidirect

Maximal subgroups

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

subgroupindex
#832072
C2×#50852
#793052
C3×#15492
#705212
#705012
#677392
#101533