Finite Group Databasebeta

← #10697874

Subgroups — #10697874

Order 768.

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

normal subgroups only

This group has 614 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#83253C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#79268C211yesmaximal, characteristic
4C3×#1546C211yesmaximal, characteristic, complemented
5#70512C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10289—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#944—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×#825—22no—
16#5088—21no—
17#5087—22no—
18#5078—21no—
19#4975—22no—
20#4975—22no—
21#4967—22no—
22C42×Dic3C2211yescharacteristic
23#4925C2211yescharacteristic
24#4846—22no—
25#4736—22no—
26#4694—21no—
27#4682—21no—
28C3×#332—22no—
29#4482—22no—
30#3226—31no—
31#3226—31no—
32#2868—31no—
33#1962—31no—
34#1546—31no—
35#1546D311yescharacteristic, complemented
36#1276—31no—
37C22×C3×Q8—21no—
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42C2311yescharacteristic
48#944—44no—
49#944—44no—
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54C2×Dic3⋊C4—28no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3D412yes—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#830—21no—
73#825—44no—
74#821—24no—
75#818—24no—
76#809—24no—
77#807—24no—
78C4⋊(C3⋊C8)D412yes—
79C4×C3⋊C8—21no—
80#532—31no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8—62no—
83C2×C4⋊Q8C2×D311yescharacteristic
84#500—61no—
85#500—61no—
86#486—61no—
87#486—61no—
88#480—62no—
89#480—62no—
90#474—62no—
91#474—62no—
92#431—31no—
93#431C2×D311yescharacteristic
94#423C2×D311yescharacteristic
95C2×#104—62no—
96#384—62no—
97#384—62no—
98C43—31no—
99#332—22no—
100#332—62no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C2×#67—41no—
110C22×Dic3—22no—
111C22×Dic3—22no—
112#235—42no—
113#235—42no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×Dic6—42no—
122C2×Dic6—42no—
123C3×M16—42no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142Dic3⋊C4—416no—
143C4×Dic3—24no—
144C4×Dic3—24no—
145C4×Dic3—22no—
146C4×Dic3D814yes—
147C2×C3⋊C8—22no—
148C2×#41—61no—
149C22×Q8—21no—
150C22×Q8—61no—
151C2×M16—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—124no—
162C4⋊Q8—124no—
163#127—31no—
164#124—61no—
165#121—61no—
166#121—61no—
167C4×D4—61no—
168C4×D4—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—32no—
171C2×C4⋊C4—68no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4D1212yes—
174C2×C42—32no—
175C2×C42—61no—
176C2×C42—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#107—64no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—32no—
183C4⋊C8#6712yes—
184#104—124no—
185#102—64no—
186#102—64no—
187C22⋊C8—21no—
188C22⋊C8—61no—
189C4×C8—21no—
190C4×C8—61no—
191C23×C3#12011yescharacteristic
192C22×D3—41no—
193C3×Q8—82no—
194C3×Q8—81no—
195C3×Q8—81no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—44no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12711yescharacteristic
206C2×C3×C4C2×D812yes—
207C2×C3×C4AGL(1,ℤ/8)12yes—
208#67—82no—
209C2×Dic3—24no—
210C2×Dic3—24no—
211C2×Dic3—24no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—44no—
215D12—42no—
216D12—42no—
217D12—42no—
218C4×D3—81no—
219C4×D3—81no—
220Dic6—42no—
221Dic6—82no—
222Dic6—81no—
223Dic6—81no—
224C3×C8—42no—
225C3×C8—81no—
226C3⋊C8—42no—
227#41—122no—
228#41—122no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—61no—
234C2×Q8—121no—
235C2×Q8—122no—
236C2×Q8—122no—
237C2×Q8—121no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C22×C4—22no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—121no—
246C22×C4C2×D1211yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249M16—42no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—62no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—1216no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C42—34no—
267C42—64no—
268C42—64no—
269C42—62no—
270C42#23511yescharacteristic
271C42D3×D411yescharacteristic
272C42D3×D411yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#39211yescharacteristic
277C22×C3#44712yes—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—82no—
284C3×C4#45612yes—
285C3×C4#45612yes—
286C3×C4#45912yes—
287C3×C4#49311yescharacteristic
288C3×C4#49611yescharacteristic
289Dic3—28no—
290Dic3—81no—
291Dic3—82no—
292C23—121no—
293C23#88211yescharacteristic
294Q8—81no—
295Q8—81no—
296Q8—82no—
297Q8—122no—
298Q8—241no—
299Q8—241no—
300Q8—242no—
301D4—122no—
302D4—122no—
303D4—122no—
304D4—242no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—64no—
310C2×C4—64no—
311C2×C4—64no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—124no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4#88211yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4D3×D812yes—
321C2×C4#91412yes—
322C2×C4#93011yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#94011yescharacteristic
325C8—42no—
326C8—81no—
327C8—122no—
328C2×C3—22no—
329C2×C3#200511yescharacteristic
330C2×C3#200511yescharacteristic
331C2×C3#201111yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#477312yes—
339C22#496011yescharacteristic
340C4—21no—
341C4—21no—
342C4—68no—
343C4—82no—
344C4—242no—
345C4—241no—
346C4#475212yes—
347C4#478612yes—
348C4#515111yescharacteristic
349C4#517611yescharacteristic
350C4#520512yes—
351C3#1028911yescharacteristic, complemented, Sylow
352C2—22no—
353C2—481no—
354C2#7011911yescharacteristic
355C2#7020911yescharacteristic
356C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1546D3
hypocenterC3#10289
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #83253C2semidirect
4. C3×#1546C2semidirect
6. #70506C2semidirect
7. #67736C2semidirect
35. #1546D3semidirect
351. C3#10289semidirect

Maximal subgroups

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

subgroupindex
#832532
#832152
#792682
C3×#15462
#705122
#705062
#677362
#102893