Finite Group Databasebeta

← #10696863

Subgroups — #10696863

Order 768.

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

normal subgroups only

This group has 506 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×#3206C211yesmaximal, characteristic, complemented
2#83306C211yesmaximal, characteristic
3C2×#5089C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71776C211yesmaximal, characteristic, complemented
6#70784C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#10136—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#492—21no—
13C3×#487—21no—
14C3×#461—22no—
15C3×#448—22no—
16C3×#431C2211yescharacteristic
17#5089—21no—
18#5089—22no—
19#5089—22no—
20#5074—21no—
21#5040—21no—
22#5036—21no—
23#5016—22no—
24C2×#810—21no—
25C2×#809—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4468—22no—
32#3213—31no—
33#3206D311yescharacteristic, complemented
34C2×#489—31no—
35#1556—31no—
36#1530—31no—
37#1469—31no—
38#1302—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×SD16—24no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#214—22no—
59C2×#213—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C22×C3⋊C8—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67C3×C4⋊C8—21no—
68C3×#103—24no—
69C3×C22⋊C8D412yes—
70C3×C4×C8—21no—
71#830—21no—
72#830—21no—
73#810—24no—
74#810—22no—
75#809—24no—
76#809—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#496—61no—
87#492—21no—
88#489—62no—
89#489—62no—
90#489—61no—
91#487—21no—
92#463—62no—
93#461—22no—
94#448—22no—
95#448—61no—
96#447—61no—
97#432—31no—
98#432—31no—
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#102—61no—
102C2×#97—31no—
103#332—62no—
104#329—62no—
105#323—62no—
106C23×D3—21no—
107C3×#41—82no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—42no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C3×SD16—84no—
129C2×C3×C8—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#214—82no—
143#213—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—42no—
151C2×C3⋊C8—41no—
152C2×#41—21no—
153C22×Q8—21no—
154C22×Q8—31no—
155C22×D4—31no—
156C2×SD16—24no—
157C2×SD16—62no—
158C2×D8—62no—
159C22×C8—61no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8#6712yes—
164C4⋊Q8#6711yescharacteristic
165#127#6711yescharacteristic
166#124—21no—
167#124—61no—
168#124—61no—
169#124—61no—
170#124—61no—
171#121—21no—
172#121—21no—
173C4×D4—21no—
174C4×D4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—32no—
182C4⋊C8—61no—
183#103—24no—
184#103—64no—
185#103—62no—
186#102—64no—
187#102—62no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190C22⋊C8#6712yes—
191#97—32no—
192#97—61no—
193#97—61no—
194#97—61no—
195C4×C8—21no—
196C23×C3—41no—
197C23×C3#12111yescharacteristic
198C22×D3—41no—
199C22×D3—42no—
200C22×D3—81no—
201C22×D3—81no—
202C3×Q8—44no—
203C3×Q8—81no—
204C3×Q8—82no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—81no—
208C3×D4—82no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—82no—
215C2×C3×C4—82no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×Dic3—41no—
223C2×Dic3—81no—
224D12—42no—
225D12—42no—
226Dic6—42no—
227Dic6—42no—
228C3×C8—42no—
229C3⋊C8—42no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C24—61no—
233#41—82no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—82no—
240C2×Q8—122no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—122no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—61no—
251C22×C4D3×D411yescharacteristic
252C22×C4#23711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254SD16—84no—
255SD16—242no—
256D8—242no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—122no—
261C2×C8—121no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42C2×#6711yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3#39211yescharacteristic
284C22×C3#47112yes—
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#45512yes—
297Dic3—81no—
298C23—41no—
299C23—121no—
300C23—122no—
301C23—241no—
302C23—241no—
303C23#93811yescharacteristic
304Q8—44no—
305Q8—81no—
306Q8—82no—
307Q8—122no—
308Q8—122no—
309D4—42no—
310D4—42no—
311D4—81no—
312D4—82no—
313D4—122no—
314D4—122no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—121no—
323C2×C4—241no—
324C2×C4#93711yescharacteristic
325C2×C4#93711yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#93911yescharacteristic
328C2×C4#94011yescharacteristic
329C2×C4#95311yescharacteristic
330C8—42no—
331C8—122no—
332C8—122no—
333C8—241no—
334C2×C3—22no—
335C2×C3—161no—
336C2×C3#200411yescharacteristic
337C2×C3#200811yescharacteristic
338C2×C3#201911yescharacteristic
339D3—161no—
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—482no—
349C22#517712yes—
350C22#522311yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—82no—
358C4—241no—
359C4#520212yes—
360C3#1013611yescharacteristic, complemented, Sylow
361C2—22no—
362C2—161no—
363C2—481no—
364C2#7185811yescharacteristic
365C2#7205911yescharacteristic
366C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radical#83306C2
hypercenter#3206D3
hypocenterC3#10136
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. C3×#3206C2semidirect
3. C2×#5089C2semidirect
4. #71798C2semidirect
5. #71776C2semidirect
6. #70784C2semidirect
7. #67742C2semidirect
33. #3206D3semidirect
360. C3#10136semidirect

Maximal subgroups

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

subgroupindex
C3×#32062
#833062
C2×#50892
#717982
#717762
#707842
#677422
#101363