Finite Group Databasebeta

← #10699210

Subgroups — #10699210

Order 768.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#79313C211yesmaximal, characteristic
2C3×#1565C211yesmaximal, characteristic
3#70456C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10505—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×#816—22no—
15C4×C4⋊Dic3C2211yescharacteristic
16#4924C2211yescharacteristic
17#4849—21no—
18#4849—21no—
19#4739—22no—
20#4699—21no—
21#4681—21no—
22C3×#335—21no—
23C3×#334—21no—
24#4555—21no—
25#4555—21no—
26#4554—21no—
27#4554—21no—
28#4482—21no—
29#4481—21no—
30#3209—31no—
31#1997—31no—
32#1986—31no—
33#1981—31no—
34#1565D311yescharacteristic
35#1550—31no—
36#1303—31no—
37C22×Dic6D412yes—
38C2×C3×M16D411yescharacteristic
39C2×C3×M16D411yescharacteristic
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46#924—21no—
47#924—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C2×Dic12—22no—
51C2×Dic12—22no—
52#870—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869D411yescharacteristic
58#869D411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D411yescharacteristic
64C3×C4⋊C8D411yescharacteristic
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—21no—
70#816—22no—
71#816—22no—
72#816—44no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#533—31no—
76C2×C4⋊Q8—31no—
77#494—61no—
78#487—61no—
79#462—61no—
80#462—61no—
81#451—62no—
82#431C2×D311yescharacteristic
83#423—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×#103—62no—
86C4×M16C2×D311yescharacteristic
87C4×C4⋊C4—31no—
88#343—61no—
89#343—61no—
90#339—61no—
91#339—61no—
92#335—21no—
93#334—21no—
94#333—61no—
95#332—61no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—22no—
100C22×Dic3—41no—
101C2×Dic6—22no—
102C2×Dic6—21no—
103C2×Dic6—22no—
104C2×Dic6—21no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C3×M16—21no—
108C3×M16—21no—
109C3×M16—21no—
110C3×M16Q1612yescomplemented
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—42no—
116C2×C3×C8—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122C4⋊Dic3—22no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—41no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137#208—41no—
138#208—41no—
139C2×C3⋊C8—41no—
140Dic12—82no—
141Dic12—82no—
142C22×Q8—32no—
143C2×Q16—62no—
144C2×Q16—62no—
145C2×M16—61no—
146C2×M16D1211yescharacteristic
147C2×M16D1211yescharacteristic
148C22×C8—21no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155#125—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160#117—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8#6711yescharacteristic
171C4⋊C8#6711yescharacteristic
172#103—62no—
173#103—62no—
174#103—124no—
175C22⋊C8—21no—
176#97—21no—
177C4×C8—21no—
178C4×C8—21no—
179#95—61no—
180C23×C3#12111yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4#12311yescharacteristic
189C2×C3×C4#12411yescharacteristic
190C2×C3×C4C2×Q1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—42no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—82no—
201Dic6—44no—
202Dic6—41no—
203Dic6—42no—
204Dic6—41no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—22no—
208C3×C8—41no—
209C3×C8—41no—
210C3×C8—41no—
211C3×C8—81no—
212C3⋊C8—81no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—122no—
218C2×Q8—122no—
219C22×C4—62no—
220C22×C4—121no—
221C22×C4C2×D1211yescharacteristic
222C22×C4D3×D411yescharacteristic
223C22×C4#23911yescharacteristic
224Q16—242no—
225Q16—242no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—121no—
230M16—121no—
231M16Dic1212yescomplemented
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—121no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C42—121no—
253C42—121no—
254C42—121no—
255C42#23511yescharacteristic
256C42D3×D411yescharacteristic
257C42#23911yescharacteristic
258C42C2×#6711yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3#39411yescharacteristic
262C22×C3#46211yescharacteristic
263C22×C3#46411yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4#45311yescharacteristic
268C3×C4#45511yescharacteristic
269C3×C4#46211yescharacteristic
270C3×C4#46411yescharacteristic
271C3×C4#46711yescharacteristic
272C3×C4#47011yescharacteristic
273Dic3—82no—
274Dic3—81no—
275Dic3—82no—
276Dic3—81no—
277C23#89511yescharacteristic
278Q8—124no—
279Q8—121no—
280Q8—122no—
281Q8—121no—
282Q8—122no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—122no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—122no—
291C2×C4—241no—
292C2×C4—241no—
293C2×C4—242no—
294C2×C4#89411yescharacteristic
295C2×C4#89411yescharacteristic
296C2×C4#89511yescharacteristic
297C2×C4#90411yescharacteristic
298C2×C4C2×Dic1211yescharacteristic
299C2×C4#90911yescharacteristic
300C2×C4#90911yescharacteristic
301C2×C4#93011yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#94711yescharacteristic
304C8—22no—
305C8—41no—
306C8—41no—
307C8—41no—
308C8—81no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203711yescharacteristic
313C2×C3#204911yescharacteristic
314C2×C3#205611yescharacteristic
315C22—21no—
316C22—21no—
317C22#484511yescharacteristic
318C22#484911yescharacteristic
319C22#499111yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—242no—
324C4—242no—
325C4—241no—
326C4—241no—
327C4#484511yescharacteristic
328C4#484911yescharacteristic
329C4#511611yescharacteristic
330C4#513711yescharacteristic
331C4#519811yescharacteristic
332C4#520211yescharacteristic
333C3#1050511yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7048711yescharacteristic
337C2#7067511yescharacteristic
338C2#7073711yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1565C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1565D3
hypocenterC3#10505
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
110. C3×M16Q16semidirect
231. M16Dic12semidirect
333. C3#10505semidirect

Maximal subgroups

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

subgroupindex
#793132
C3×#15652
#704562
#704382
#698792
#698682
#677462
#105053