Finite Group Databasebeta

← #10696041

Subgroups — #10696041

Order 768.

566 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 99 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#82467C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3#79309C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70513C211yesmaximal, characteristic
6#70500C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10014—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5230—21no—
15#5227—21no—
16C2×#825—22no—
17#5027—21no—
18C4×#215—22no—
19#5020—21no—
20#4983—21no—
21#4981—21no—
22#4979—21no—
23#4968—21no—
24#4932C2211yescharacteristic
25C4×Dic3⋊C4—22no—
26#4925C2211yescharacteristic
27#4851—21no—
28#4848—21no—
29#4738—21no—
30#4735—21no—
31C4×#204—22no—
32C3×Q8⋊C8—22no—
33#4484—22no—
34#3103—31no—
35#3103—31no—
36#2399—31no—
37#1963—31no—
38#1487—31no—
39#1487D311yescharacteristic, complemented
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D412yes—
62C2×#204—22no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#825—44no—
84#821—22no—
85#817—22no—
86#816—22no—
87#809—22no—
88#808—22no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532—31no—
92C2×C4×Q8—31no—
93C2×C4×Q8C2×D311yescharacteristic
94#481—61no—
95#481—61no—
96#475—61no—
97#475—61no—
98#460—61no—
99#460—61no—
100#448—61no—
101#448—61no—
102C4×SD16—62no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107C2×#104—62no—
108#398—61no—
109#398—61no—
110#395—61no—
111#395—61no—
112C4×C4⋊C4—62no—
113#376—31no—
114Q8⋊C8—22no—
115Q8⋊C8—62no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—41no—
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128#235—42no—
129#235—42no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—42no—
138C3×M16—42no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#215—44no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—44no—
159Dic3⋊C4—44no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—22no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C4×Dic3SD1614yescomplemented
166C2×C3⋊C8—22no—
167#204—44no—
168C2×#41—61no—
169C22×Q8—31no—
170C22×Q8D1211yescharacteristic
171C2×SD16—62no—
172C2×SD16—62no—
173C2×M16—21no—
174C4⋊Q8—31no—
175#127—31no—
176#124—61no—
177#121—61no—
178#121—61no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C42—21no—
196C2×C42—32no—
197C2×C42—61no—
198C2×C42C22×D311yescharacteristic
199#106—62no—
200#106—62no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—61no—
205#104—124no—
206#103—62no—
207#103—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—32no—
213C4×C8#6712yes—
214#95—62no—
215#95—62no—
216#95—61no—
217#95—61no—
218C23×C3#12011yescharacteristic
219C22×D3—41no—
220C3×Q8—22no—
221C3×Q8—22no—
222C3×Q8—41no—
223C3×Q8—41no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237C2×C3×C4C2×SD1612yes—
238C2×C3×C4#13712yes—
239#67—82no—
240C2×Dic3—22no—
241C2×Dic3—24no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249D12—42no—
250D12—42no—
251D12—42no—
252C4×D3—81no—
253C4×D3—81no—
254Dic6—22no—
255Dic6—22no—
256Dic6—41no—
257Dic6—41no—
258Dic6—42no—
259C3×C8—42no—
260C3×C8—81no—
261C3⋊C8—42no—
262#41—122no—
263#41—122no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—42no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—122no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—121no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—62no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—121no—
281C22×C4C2×D1211yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284SD16—124no—
285SD16—124no—
286M16—42no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—62no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—124no—
299C4⋊C4—124no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—34no—
305C42—41no—
306C42—41no—
307C42—62no—
308C42—121no—
309C42—121no—
310C42#23511yescharacteristic
311C42D3×D411yescharacteristic
312C42D3×D411yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3#39211yescharacteristic
317C22×C3#44812yes—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—81no—
327C3×C4#46112yes—
328C3×C4#49211yescharacteristic
329C3×C4#49711yescharacteristic
330Dic3—44no—
331Dic3—42no—
332Dic3—81no—
333Dic3—81no—
334C23—121no—
335C23#88211yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—41no—
339Q8—41no—
340Q8—62no—
341Q8—62no—
342Q8—121no—
343Q8—121no—
344Q8—122no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—242no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#88211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#88411yescharacteristic
370C2×C4D3×SD1612yes—
371C2×C4#91812yes—
372C2×C4#93011yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#94011yescharacteristic
375C8—42no—
376C8—81no—
377C8—122no—
378C2×C3—22no—
379C2×C3#201011yescharacteristic
380C2×C3#201011yescharacteristic
381C2×C3#201311yescharacteristic
382D3—161no—
383C22—21no—
384C22—21no—
385C22—241no—
386C22—241no—
387C22—241no—
388C22#480612yes—
389C22#496011yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—124no—
397C4—122no—
398C4—241no—
399C4—241no—
400C4#481812yes—
401C4#517511yescharacteristic
402C4#519011yescharacteristic
403C3#1001411yescharacteristic, complemented, Sylow
404C2—22no—
405C2—481no—
406C2#6999311yescharacteristic
407C2#7028211yescharacteristic
408C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1487D3
hypocenterC3#10014
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
1. #82467C2semidirect
4. C3×#1487C2semidirect
6. #70500C2semidirect
7. #67737C2semidirect
39. #1487D3semidirect
165. C4×Dic3SD16semidirect
403. C3#10014semidirect

Maximal subgroups

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

subgroupindex
#824672
#824422
#793092
C3×#14872
#705132
#705002
#677372
#100143