Finite Group Databasebeta

← #10696570

Subgroups — #10696570

Order 768.

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

normal subgroups only

This group has 590 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#83306C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#73432C211yesmaximal, characteristic, complemented
4C3×#1505C211yesmaximal, characteristic, complemented
5#70822C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10094—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#875—21no—
11C2×#875C2211yescharacteristic
12C2×#869—21no—
13C2×#869C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#376C2211yescharacteristic
16#5232—22no—
17#5089—21no—
18#5079—21no—
19#5074—21no—
20#5070—21no—
21#5055—22no—
22#5040—21no—
23#5036—21no—
24#5016—22no—
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4677—22no—
28#4677—22no—
29C3×#327—22no—
30#4486—22no—
31#4464—22no—
32#3213—31no—
33#3141—31no—
34#2658—31no—
35#1556—31no—
36#1527—31no—
37#1505D311yescharacteristic, complemented
38#1300—31no—
39C2×C3×#41—21no—
40C22×D12D411yescharacteristic
41C22×Dic6—21no—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—22no—
53C2×#213—22no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D412yes—
60C2×#208—21no—
61#875—21no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—44no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—44no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#95—21no—
75C3×#95D412yes—
76#835—24no—
77#832—24no—
78#830—21no—
79#810—22no—
80#809—22no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×C4⋊Q8—61no—
88C2×#124—31no—
89C2×#124—61no—
90#496—61no—
91#489—62no—
92#489—61no—
93#487—61no—
94#486—61no—
95#463—62no—
96#448—61no—
97#447—61no—
98#432—31no—
99#432C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102#399—62no—
103#399—62no—
104#376C2×D311yescharacteristic
105#332—62no—
106#329—62no—
107#327—22no—
108C23×D3—21no—
109C3×#41—44no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—22no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—42no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—24no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#214—82no—
143#213—82no—
144(C2×D3)⋊C4—48no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—48no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154#208—42no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×#41—21no—
158C22×Q8—31no—
159C22×Q8—61no—
160C22×D4—31no—
161C2×SD16—62no—
162C2×D8—62no—
163C2×M16—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—124no—
169C4⋊Q8D1211yescharacteristic
170#127D1211yescharacteristic
171#124—21no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176#124—124no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—32no—
191#103—62no—
192#103—64no—
193#102—62no—
194#102—64no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—61no—
200C4×C8—61no—
201#95—21no—
202#95#6712yes—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—41no—
208C22×D3—42no—
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—81no—
213C3×D4—82no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—24no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—82no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233D12—42no—
234D12—81no—
235Dic6—42no—
236Dic6—82no—
237Dic6—82no—
238Dic6—81no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C24—61no—
243#41—44no—
244C2×Q8—21no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—122no—
248C2×Q8—122no—
249C2×Q8—122no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—122no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—62no—
262C22×C4—61no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—242no—
267D8—242no—
268M16—122no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—62no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—128no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—128no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C42—24no—
287C42C2×D1211yescharacteristic
288C42#23511yescharacteristic
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3#39211yescharacteristic
297C22×C3#45412yes—
298C2×D3—84no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—44no—
308C3×C4—81no—
309C3×C4#45312yes—
310Dic3—81no—
311Dic3—82no—
312C23—41no—
313C23—122no—
314C23—122no—
315C23—122no—
316C23—121no—
317C23#93711yescharacteristic
318Q8—42no—
319Q8—122no—
320Q8—241no—
321Q8—242no—
322Q8—242no—
323D4—42no—
324D4—42no—
325D4—81no—
326D4—82no—
327D4—122no—
328D4—241no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4#88211yescharacteristic
343C2×C4#88211yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4#93011yescharacteristic
347C2×C4#94011yescharacteristic
348C8—81no—
349C8—122no—
350C8—122no—
351C2×C3—22no—
352C2×C3—161no—
353C2×C3#200211yescharacteristic
354C2×C3#200311yescharacteristic
355C2×C3#200911yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—244no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22#496011yescharacteristic
367C22#520112yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—81no—
375C4—241no—
376C4—242no—
377C4#482212yes—
378C3#1009411yescharacteristic, complemented, Sylow
379C2—22no—
380C2—161no—
381C2—481no—
382C2#7005311yescharacteristic
383C2#7032611yescharacteristic
384C2#7038511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#392
prime radical#83306C2
hypercenter#1505D3
hypocenterC3#10094
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #73432C2semidirect
4. C3×#1505C2semidirect
5. #70822C2semidirect
6. #70804C2semidirect
7. #67615C2semidirect
37. #1505D3semidirect
378. C3#10094semidirect

Maximal subgroups

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

subgroupindex
#833062
#832832
#734322
C3×#15052
#708222
#708042
#676152
#100943