Finite Group Databasebeta

← #10696842

Subgroups — #10696842

Order 768.

536 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3230C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3C2×#5086C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic, complemented
5#71776C211yesmaximal, characteristic, complemented
6#70784C211yesmaximal, characteristic
7#67742C211yesmaximal, characteristic, complemented
8#10133—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C22×#216—21no—
12C22×#215—21no—
13C2×#875C2211yescharacteristic
14C3×#500—21no—
15C3×#487—21no—
16C3×#484—22no—
17C3×#477—22no—
18C3×#431C2211yescharacteristic
19#5089—21no—
20#5086—22no—
21#5086—21no—
22#5086—22no—
23#5050—21no—
24#5046—21no—
25#5028—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4547—22no—
30#4543—22no—
31#4468—22no—
32#3230D311yescharacteristic, complemented
33#3215—31no—
34C2×#497—31no—
35#1556—31no—
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—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#216—24no—
58C2×#216—22no—
59C2×#216—44no—
60C2×#215—24no—
61C2×#215—22no—
62C2×#215—44no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C22×C3⋊C8—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70C3×#107—24no—
71C3×C4⋊C8—21no—
72C3×#103—24no—
73C3×C22⋊C8D412yes—
74C3×C4×C8—21no—
75#830—21no—
76#830—21no—
77#810—22no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#803—21no—
84#803D412yes—
85C22×Q16—61no—
86C22×SD16—61no—
87#533C2×D311yescharacteristic
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×#124—31no—
90#500—21no—
91#497—62no—
92#497—62no—
93#497—61no—
94#489—61no—
95#487—21no—
96#484—22no—
97#477—22no—
98#464—62no—
99#451—61no—
100#450—61no—
101#432—31no—
102#432—31no—
103#431C2×D311yescharacteristic
104C2×#97—31no—
105#332—62no—
106#330—62no—
107#323—62no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—42no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#216—48no—
144#216—82no—
145#215—48no—
146#215—82no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C2×C3⋊C8—24no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C22×Q8—21no—
156C22×Q8—31no—
157C22×D4—31no—
158C2×Q16—64no—
159C2×Q16—62no—
160C2×Q16—124no—
161C2×SD16—64no—
162C2×SD16—62no—
163C2×SD16—124no—
164C22×C8—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8#6712yes—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42C22×D311yescharacteristic
186#107—24no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190#103—24no—
191#103—62no—
192#102—62no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8#6712yes—
196#97—32no—
197#97—61no—
198#97—61no—
199#97—61no—
200C4×C8—21no—
201C23×C3#12111yescharacteristic
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—81no—
205C22×D3—81no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—82no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—82no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—81no—
226D12—42no—
227D12—42no—
228Dic6—42no—
229Dic6—42no—
230C3×C8—42no—
231C3⋊C8—24no—
232C3⋊C8—42no—
233C3⋊C8—81no—
234C24—61no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—21no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
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×C4D3×D411yescharacteristic
250C22×C4#23711yescharacteristic
251C22×C4C2×#6711yescharacteristic
252Q16—128no—
253Q16—242no—
254SD16—128no—
255SD16—242no—
256C2×C8—22no—
257C2×C8—64no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—41no—
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42C2×#6711yescharacteristic
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—81no—
296C3×C4—81no—
297C3×C4#45312yes—
298Dic3—81no—
299C23—121no—
300C23—122no—
301C23—241no—
302C23—241no—
303C23#93811yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308Q8—82no—
309Q8—122no—
310Q8—122no—
311D4—122no—
312D4—122no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—121no—
321C2×C4—241no—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93911yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—64no—
330C8—122no—
331C8—241no—
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#519212yes—
344C22#522311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4—241no—
354C4#519812yes—
355C3#1013311yescharacteristic, complemented, Sylow
356C2—22no—
357C2—481no—
358C2#7186911yescharacteristic
359C2#7207411yescharacteristic
360C2#7209411yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3230C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3230D3
hypocenterC3#10133
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
1. C3×#3230C2semidirect
4. #71822C2semidirect
5. #71776C2semidirect
7. #67742C2semidirect
32. #3230D3semidirect
355. C3#10133semidirect

Maximal subgroups

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

subgroupindex
C3×#32302
#833572
C2×#50862
#718222
#717762
#707842
#677422
#101333