Finite Group Databasebeta

← #10696845

Subgroups — #10696845

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#82949C211yesmaximal, characteristic
2C2×#5060C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic, complemented
4C3×#1469C211yesmaximal, characteristic
5#70822C211yesmaximal, characteristic, complemented
6#70813C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#10133—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C22×#216—21no—
11C22×#214—21no—
12#5470C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5228—21no—
17#5218—21no—
18#5060—21no—
19#5060—22no—
20#5060—22no—
21#5055—21no—
22#5054—22no—
23#4926C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4850—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29C3×#323—22no—
30#4486—22no—
31#4465—22no—
32#3230—31no—
33#3215—31no—
34C2×#497—31no—
35#1556—31no—
36#1531—31no—
37#1469D311yescharacteristic
38#1302—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—24no—
50C2×#216—22no—
51C2×#216—44no—
52C2×#214—24no—
53C2×#214—22no—
54C2×#214—44no—
55#926—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58#924—21no—
59#924—21no—
60C22×C3⋊C8—21no—
61#870—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67#869D412yes—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×#97—21no—
75C3×C4×C8—21no—
76#835—22no—
77#832—22no—
78#830—21no—
79#818—24no—
80#816—24no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#803—21no—
84#803D412yes—
85C22×Q16—61no—
86C22×SD16—61no—
87#533—31no—
88C2×C4⋊Q8—31no—
89C2×#124C2×D311yescharacteristic
90#500—61no—
91#497—62no—
92#497—62no—
93#497—61no—
94#489—61no—
95#487—61no—
96#484—62no—
97#477—62no—
98#464—62no—
99#451—61no—
100#450—61no—
101#432—31no—
102#432C2×D311yescharacteristic
103#431C2×D311yescharacteristic
104C2×#97—31no—
105#332—62no—
106#330—62no—
107#323—22no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137#216—48no—
138#216—82no—
139#214—48no—
140#214—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152C2×C3⋊C8—24no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C22×Q8—61no—
156C22×Q8D1211yescharacteristic
157C22×D4D1211yescharacteristic
158C2×Q16—64no—
159C2×Q16—62no—
160C2×Q16—124no—
161C2×SD16—64no—
162C2×SD16—62no—
163C2×SD16—124no—
164C22×C8—61no—
165C4⋊Q8—32no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171#125—61no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180#117—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—32no—
190#103—64no—
191#103—62no—
192#102—62no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C22⋊C8#6712yes—
196#97—21no—
197#97—32no—
198#97—61no—
199#97—61no—
200C4×C8—21no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3#12111yescharacteristic
206C3×Q8—42no—
207C3×Q8—42no—
208C3×D4—42no—
209C3×D4—42no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—81no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—82no—
225Dic6—42no—
226Dic6—42no—
227Dic6—42no—
228Dic6—42no—
229Dic6—82no—
230C3×C8—42no—
231C3×C8—81no—
232C3⋊C8—24no—
233C3⋊C8—42no—
234C24—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—42no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—61no—
241C2×Q8—122no—
242C2×Q8—122no—
243C2×Q8—121no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—121no—
249C22×C4C2×D1211yescharacteristic
250C22×C4#23511yescharacteristic
251C22×C4C2×#6711yescharacteristic
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—61no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—121no—
276C42—121no—
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—162no—
287C22×C3#39211yescharacteristic
288C22×C3#47312yes—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—81no—
295C3×C4#45312yes—
296Dic3—82no—
297Dic3—81no—
298Dic3—81no—
299C23—41no—
300C23—42no—
301C23—81no—
302C23—81no—
303C23#93011yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—122no—
307Q8—122no—
308Q8—122no—
309Q8—122no—
310Q8—242no—
311D4—42no—
312D4—42no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—122no—
321C2×C4—242no—
322C2×C4#88211yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#88411yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#94011yescharacteristic
328C8—42no—
329C8—64no—
330C8—81no—
331C8—122no—
332C2×C3—22no—
333C2×C3—161no—
334C2×C3#200211yescharacteristic
335C2×C3#200911yescharacteristic
336C2×C3#202011yescharacteristic
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—162no—
343C22#496011yescharacteristic
344C22#514312yes—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—81no—
351C4—241no—
352C4—242no—
353C4—241no—
354C4#474712yes—
355C3#1013311yescharacteristic, complemented, Sylow
356C2—22no—
357C2—161no—
358C2#7000811yescharacteristic
359C2#7005311yescharacteristic
360C2#7008211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1469C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1469D3
hypocenterC3#10133
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
3. #79272C2semidirect
5. #70822C2semidirect
6. #70813C2semidirect
7. #67742C2semidirect
355. C3#10133semidirect

Maximal subgroups

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

subgroupindex
#829492
C2×#50602
#792722
C3×#14692
#708222
#708132
#677422
#101333