Finite Group Databasebeta

← #10696843

Subgroups — #10696843

Order 768.

536 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 83 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#83224C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic
3#79351C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic, complemented
5#70828C211yesmaximal, characteristic
6#70813C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#10133—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C22×#215—21no—
12C2×#875C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#432C2211yescharacteristic
16#5093—21no—
17#5091—22no—
18#5086—22no—
19#5086—22no—
20#5078—21no—
21#4970—22no—
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4845—22no—
25#4739—21no—
26#4731—21no—
27#4715—21no—
28#4714—21no—
29C3×#332—22no—
30#4468—22no—
31#4465—22no—
32#3230—31no—
33#3215—31no—
34C2×#497—31no—
35#1556D311yescharacteristic, complemented
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
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×#216—24no—
50C2×#216—44no—
51C2×#215—24no—
52C2×#215—44no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55#926—21no—
56#924—21no—
57#924—21no—
58C22×C3⋊C8—21no—
59C2×Dic12—22no—
60C2×#204—22no—
61#875—21no—
62#875—21no—
63#875—21no—
64#875—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75C3×#97—21no—
76#830D412yes—
77#821—22no—
78#816—22no—
79#810—24no—
80#807—24no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85C22×Q16—61no—
86C22×SD16—61no—
87#533—31no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×#124—31no—
90#500—61no—
91#497—61no—
92#497—62no—
93#497—62no—
94#489—61no—
95#487—61no—
96#484—62no—
97#477—62no—
98#464—62no—
99#451—61no—
100#450—61no—
101#432C2×D311yescharacteristic
102#432C2×D311yescharacteristic
103#431—31no—
104C2×#97—31no—
105#332—22no—
106#330—62no—
107#323—62no—
108C23×D3—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—42no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—42no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137#216—48no—
138#215—48no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C2×C3⋊C8—24no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153Dic12—82no—
154#204—82no—
155C22×Q8—21no—
156C22×Q8—31no—
157C22×D4—31no—
158C2×Q16—62no—
159C2×Q16—64no—
160C2×Q16—124no—
161C2×SD16—62no—
162C2×SD16—64no—
163C2×SD16—124no—
164C22×C8—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170C4⋊Q8D1212yes—
171#125—61no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—64no—
191#103—62no—
192#102—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—32no—
196#97—21no—
197#97—21no—
198#97—32no—
199#97—61no—
200C4×C8—61no—
201C23×C3#12111yescharacteristic
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—81no—
205C22×D3—81no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—82no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—82no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—81no—
224D12—42no—
225D12—42no—
226Dic6—42no—
227Dic6—42no—
228Dic6—42no—
229Dic6—42no—
230C3×C8—42no—
231C3×C8—81no—
232C3⋊C8—24no—
233C3⋊C8—42no—
234C24—61no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—122no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—122no—
247C22×C4—41no—
248C22×C4—61no—
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252Q16—128no—
253Q16—242no—
254SD16—128no—
255SD16—242no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—64no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—121no—
277C42C2×D1211yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#39211yescharacteristic
284C22×C3#47312yes—
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C2×D3—162no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—82no—
295C3×C4#45312yes—
296Dic3—81no—
297Dic3—81no—
298Dic3—81no—
299C23—121no—
300C23—122no—
301C23—241no—
302C23—241no—
303C23#88411yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—82no—
307Q8—122no—
308Q8—122no—
309Q8—122no—
310Q8—122no—
311D4—122no—
312D4—122no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—241no—
322C2×C4#88211yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#93011yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#94011yescharacteristic
328C8—42no—
329C8—64no—
330C8—81no—
331C8—122no—
332C2×C3—22no—
333C2×C3#200211yescharacteristic
334C2×C3#200911yescharacteristic
335C2×C3#202011yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22—482no—
343C22#475512yes—
344C22#496011yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—82no—
351C4—241no—
352C4—241no—
353C4—241no—
354C4#519812yes—
355C3#1013311yescharacteristic, complemented, Sylow
356C2—22no—
357C2—481no—
358C2#7012511yescharacteristic
359C2#7032411yescharacteristic
360C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1556D3
hypocenterC3#10133
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
1. #83224C2semidirect
4. C3×#1556C2semidirect
6. #70813C2semidirect
7. #67751C2semidirect
35. #1556D3semidirect
355. C3#10133semidirect

Maximal subgroups

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

subgroupindex
#832242
C2×#50862
#793512
C3×#15562
#708282
#708132
#677512
#101333