Finite Group Databasebeta

← #11779057

Subgroups — #11779057

Order 1152.

520 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 104 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×Q8×Dic6C211yesmaximal, characteristic, complemented
2#10592726C211yesmaximal, characteristic
3#10590373C211yesmaximal, characteristic
4#10590367C211yesmaximal, characteristic
5C3×#4535C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589084C211yesmaximal, characteristic, complemented
8#71939—31nomaximal
9#68264—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C2×#3755—21no—
12#64083C2211yescharacteristic
13C3×Q8×Dic3—21no—
14C3×#888—22no—
15C3×#888—21no—
16C3×#869—21no—
17C3×#869C2211yescharacteristic
18C3×C4×Dic6—21no—
19C3×C4×Dic6C2211yescharacteristic
20C2×#3717—21no—
21C2×#3717—22no—
22#63636—21no—
23C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25C3×C4×C3⋊C8C2211yescharacteristic
26#63564—21no—
27C3×Q82—31no—
28Q8×Dic6D311yescharacteristic, complemented
29#5085—31no—
30#5027—31no—
31#5027—31no—
32#5027—31no—
33#5027—61no—
34#4848—31no—
35#4693—31no—
36C3×Q8⋊C8—31no—
37#4546—31no—
38#4535—31no—
39#4535D311yescharacteristic, complemented
40#4481—31no—
41#4456—31no—
42C2×C32×Q8D411yescharacteristic
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C3×Dic6—22no—
47C2×C3×Dic6D411yescharacteristic
48C2×C3×Dic6D412yes—
49C32×C4⋊C4—21no—
50C32×C4⋊C4D411yescharacteristic
51C32×C42—21no—
52C32×C42C2311yescharacteristic
53#3755—42no—
54C2×C3⋊(C3⋊C8)—21no—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3D411yescharacteristic
59C3×Dic3⋊C4—22no—
60C3×Dic3⋊C4—21no—
61C3×C4×Dic3—22no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65#3717—44no—
66#3717—42no—
67#1623—91noSylow
68C3×C4⋊Q8—31no—
69C3×C4⋊Q8—61no—
70C3×C4⋊Q8—62no—
71C3×C4⋊Q8—61no—
72C3×C4×Q8—21no—
73C3×C4×Q8—31no—
74C3×C4×Q8—61no—
75C3×C4×Q8—61no—
76C3×C4×Q8C2×D311yescharacteristic
77C3×C4×Q8C2×D311yescharacteristic
78Q8×Dic3—21no—
79C2×#215—61no—
80C2×#215—62no—
81C2×#215—62no—
82C2×#215—61no—
83C2×#215—61no—
84C2×#204—62no—
85C2×#204—61no—
86#888—22no—
87#888—21no—
88#874—31no—
89#874—31no—
90#874—61no—
91#869—21no—
92#869C2×D311yescharacteristic
93C4×Dic6—21no—
94C4×Dic6C2×D311yescharacteristic
95C3×C4⋊C8—31no—
96C3×C4×C8—31no—
97#821—61no—
98#809—61no—
99#809—62no—
100#809—61no—
101#809—61no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3⋊D3—41no—
108C32×Q8—21no—
109C32×Q8SD1612yescomplemented
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C4⋊(C3⋊D3)—41no—
116C4⋊(C3⋊D3)—41no—
117C4⋊(C3⋊D3)—42no—
118C4⋊(C3⋊D3)—41no—
119C4⋊(C3⋊D3)—41no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—21no—
123C3×Dic6—21no—
124C3×Dic6—42no—
125C3×Dic6—82no—
126C3×Dic6SD1612yescomplemented
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—41no—
130Q82—31no—
131#492—91no—
132#460—91no—
133#460—91no—
134#333—91no—
135Q8⋊C8—31no—
136Q8⋊C8—91no—
137C2×C3×Q8—21no—
138C2×C3×Q8—31no—
139C2×C3×Q8—32no—
140C2×C3×Q8—62no—
141C2×C3×Q8D1211yescharacteristic
142C2×C3×Q8#6711yescharacteristic
143C2×D12—62no—
144C2×D12—61no—
145C2×D12—61no—
146C2×D12—61no—
147C2×D12—61no—
148C2×D12—61no—
149C2×D12—61no—
150C2×D12—121no—
151C2×D12—121no—
152C2×Dic6—22no—
153C2×Dic6#6711yescharacteristic
154C2×Dic6#6712yes—
155C2×C3×C8—61no—
156C2×C3×C8—61no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4D1211yescharacteristic
168C3×C4⋊C4#6711yescharacteristic
169C3×C42—21no—
170C3×C42—21no—
171C3×C42—21no—
172C3×C42—41no—
173C3×C42—62no—
174C3×C42—61no—
175C3×C42C22×D311yescharacteristic
176C3×C42C22×D311yescharacteristic
177#215—124no—
178#215—122no—
179#215—124no—
180#215—122no—
181#215—122no—
182C4⋊Dic3—22no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3#6711yescharacteristic
186Dic3⋊C4—22no—
187Dic3⋊C4—21no—
188C4×Dic3—22no—
189C4×Dic3—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—62no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—61no—
195#204—124no—
196#204—122no—
197C22×C32#12011yescharacteristic
198C2×C3⋊D3—81no—
199C2×C3⋊D3—81no—
200C2×C3⋊D3—81no—
201C32×C4—22no—
202C32×C4—21no—
203C32×C4—41no—
204C32×C4C2×SD1611yescharacteristic
205C32×C4C2×SD1611yescharacteristic
206C32×C4#13711yescharacteristic
207C32×C4#13711yescharacteristic
208C3×Dic3—22no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C2×SD16—181no—
212C2×SD16—182no—
213C2×SD16—181no—
214C4⋊Q8—31no—
215C4⋊Q8—61no—
216C4⋊Q8—62no—
217C4⋊Q8—61no—
218#127—91no—
219C4×Q8—31no—
220C4×Q8—61no—
221C4×Q8—61no—
222C4×Q8D3211yescharacteristic
223C4⋊C8—31no—
224C4⋊C8—91no—
225#102—181no—
226#102—181no—
227C4×C8—31no—
228C22×D3—122no—
229C22×D3—121no—
230C22×D3—121no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—22no—
234C3×Q8—32no—
235C3×Q8—41no—
236C3×Q8—61no—
237C3×Q8—122no—
238C3×Q8—242no—
239C3×Q8#20412yescomplemented
240C3×Q8#21512yescomplemented
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4C2×D1211yescharacteristic
254C2×C3×C4D3×D411yescharacteristic
255C2×C3×C4D3×D411yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×C3×C4C2×#6711yescharacteristic
258C2×C3×C4C2×#6711yescharacteristic
259C2×Dic3—21no—
260C2×Dic3—22no—
261C2×Dic3—21no—
262D12—122no—
263D12—122no—
264D12—122no—
265D12—122no—
266D12—121no—
267D12—121no—
268D12—121no—
269D12—121no—
270D12—121no—
271D12—121no—
272D12—121no—
273D12—121no—
274D12—241no—
275D12—241no—
276D12—242no—
277Dic6—21no—
278Dic6—42no—
279Dic6—82no—
280Dic6#21512yescomplemented
281C3×C8—62no—
282C3×C8—121no—
283C3⋊C8—22no—
284C3⋊C8—41no—
285C3⋊C8—122no—
286C3⋊C8—121no—
287C3⋊C8—121no—
288C2×C32#44811yescharacteristic
289C2×C32#44811yescharacteristic
290C2×C32#45311yescharacteristic
291C3⋊D3—161no—
292C2×Q8—31no—
293C2×Q8—32no—
294C2×Q8—62no—
295C2×Q8#62611yescharacteristic
296C2×D4—181no—
297C2×D4—181no—
298C2×D4—181no—
299SD16—364no—
300SD16—362no—
301SD16—362no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—181no—
305C4⋊C4—21no—
306C4⋊C4—31no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4#62611yescharacteristic
313C42—21no—
314C42—62no—
315C42—61no—
316C42C2×D3211yescharacteristic
317C22×C3—21no—
318C22×C3#88211yescharacteristic
319C22×C3#95311yescharacteristic
320C2×D3—242no—
321C2×D3—242no—
322C2×D3—242no—
323C2×D3—241no—
324C2×D3—241no—
325C2×D3—241no—
326C2×D3—241no—
327C2×D3—241no—
328C2×D3—241no—
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—41no—
338C3×C4—42no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—62no—
342C3×C4—81no—
343C3×C4—122no—
344C3×C4—121no—
345C3×C4C2×#20411yescharacteristic
346C3×C4#90911yescharacteristic
347C3×C4D3×SD1611yescharacteristic
348C3×C4#91811yescharacteristic
349C3×C4C2×#21511yescharacteristic
350C3×C4C2×#21511yescharacteristic
351C3×C4#94211yescharacteristic
352C3×C4#94211yescharacteristic
353Dic3—22no—
354Dic3—42no—
355Dic3—41no—
356C32#162311yescharacteristic, complemented, Sylow
357C23—361no—
358Q8—21no—
359Q8—32no—
360Q8—61no—
361Q8—122no—
362Q8—242no—
363Q8#371712yescomplemented
364D4—362no—
365D4—361no—
366D4—361no—
367D4—361no—
368D4—361no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—62no—
374C2×C4C2×#62611yescharacteristic
375C2×C4D3×#6711yescharacteristic
376C2×C4D3×#6711yescharacteristic
377C8—62no—
378C8—121no—
379C8—361no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3#473811yescharacteristic
384C2×C3#480611yescharacteristic
385C2×C3#482211yescharacteristic
386C2×C3#522711yescharacteristic
387C2×C3#522711yescharacteristic
388C2×C3#524111yescharacteristic
389D3—482no—
390D3—481no—
391D3—481no—
392C22—721no—
393C22—721no—
394C22—721no—
395C22#6397711yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—62no—
400C4—122no—
401C4—121no—
402C4C2×#371711yescharacteristic
403C4#6383311yescharacteristic
404C4D3×#21511yescharacteristic
405C4#6394811yescharacteristic
406C3—21no—
407C3#6826411yescharacteristic, complemented
408C3#7193911yescharacteristic, complemented
409C2—1441no—
410C2#1059117411yescharacteristic
411C2#1059158411yescharacteristic
412C2#1059166611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radical#64083C22
hypercenterC4×Q8D32
hypocenterC32#1623
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×Dic6C2semidirect
5. C3×#4535C2semidirect
6. #10589095C2semidirect
7. #10589084C2semidirect
28. Q8×Dic6D3semidirect
39. #4535D3semidirect
109. C32×Q8SD16semidirect
126. C3×Dic6SD16semidirect
239. C3×Q8#204semidirect
240. C3×Q8#215semidirect
280. Dic6#215semidirect
356. C32#1623semidirect
363. Q8#3717semidirect
407. C3#68264semidirect
408. C3#71939semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×Dic62
#105927262
#105903732
#105903672
C3×#45352
#105890952
#105890842
#719393
#682643