Finite Group Databasebeta

← #11781677

Subgroups — #11781677

Order 1152.

384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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#10592793C211yesmaximal, characteristic, complemented
2C3×#5085C211yesmaximal, characteristic
3#10590377C211yesmaximal, characteristic
4#10590296C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71273—31nomaximal
9#68472—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×#634C2211yescharacteristic
12C2×C3×#215—21no—
13C2×C3×#215—21no—
14C3×#874C2211yescharacteristic
15C2×#3716—21no—
16#63637—21no—
17#63635—21no—
18C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63572—21no—
23#63563—21no—
24#63555—21no—
25C3×#492—31no—
26#5094—31no—
27#5094—31no—
28#5094—61no—
29#5085D311yescharacteristic
30#5075—31no—
31#4848—31no—
32#4831—31no—
33#4679—31no—
34C3×#332—31no—
35#4547D311yescharacteristic, complemented
36#4544—31no—
37#4485—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×#634D411yescharacteristic
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C42C2311yescharacteristic
48C4⋊(C3⋊Dic3)D411yescharacteristic
49#3750—21no—
50C4×C3⋊Dic3—21no—
51C2×C3⋊(C3⋊C8)—21no—
52C3×#215—42no—
53C3×#215—42no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#3716—42no—
59#1686—91noSylow
60C2×C3×SD16—61no—
61C2×C3×SD16—61no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×#127—31no—
66C2×#215—21no—
67C2×#215—21no—
68C2×#214—61no—
69C2×#204—61no—
70#874C2×D311yescharacteristic
71C4×Dic6—31no—
72C4×Dic6—31no—
73C4×Dic6—61no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8—31no—
76C3×C4×C8—31no—
77#832—61no—
78#821—61no—
79#815—61no—
80#815—61no—
81Dic3⋊C8—61no—
82Dic3⋊C8—61no—
83#810—61no—
84#810—61no—
85#810—62no—
86#808—61no—
87#808—61no—
88#808—62no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—61no—
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4⋊(C3⋊C8)C2×D311yescharacteristic
94C4×C3⋊C8C2×D311yescharacteristic
95C22×C3×D3—41no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104#634—21no—
105#634—21no—
106C3×D12—41no—
107C3×D12—41no—
108C3×D12—41no—
109C3×D12—41no—
110C3×D12—81no—
111C3⋊(C3⋊C8)—41no—
112C3×C3⋊C8—21no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115C3×C3⋊C8SD1612yescomplemented
116#492—31no—
117#475—91no—
118#460—91no—
119#446—91no—
120#335—91no—
121#332—31no—
122#331—91no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—41no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C2×C3×D4—61no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×Dic6—31no—
133C2×Dic6—31no—
134C2×Dic6—61no—
135C3×SD16—122no—
136C3×SD16—122no—
137C2×C3×C8—31no—
138C2×C3×C8—31no—
139C2×C3×C8—61no—
140C2×C3×C8—61no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C42—21no—
148C3×C42C22×D311yescharacteristic
149C3×C42C22×D311yescharacteristic
150#215—42no—
151#215—42no—
152#214—122no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—121no—
159C4×Dic3—61no—
160C4×Dic3—61no—
161C4×Dic3—121no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—62no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8—61no—
167C2×C3⋊C8#6711yescharacteristic
168C2×C3⋊C8#6711yescharacteristic
169#204—122no—
170C22×C32#12111yescharacteristic
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—21no—
175C32×C4—81no—
176C32×C4C2×SD1611yescharacteristic
177C32×C4AGL(1,ℤ/8)11yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C32×C4#13711yescharacteristic
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C2×SD16—61no—
183C2×SD16—61no—
184C2×SD16—181no—
185C4⋊Q8D3211yescharacteristic
186#127—31no—
187C4×Q8—91no—
188#106—181no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—91no—
192C4⋊C8—181no—
193#103—181no—
194#102—181no—
195#97—181no—
196C4×C8—31no—
197C23×C3—121no—
198C22×D3—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—81no—
204C3×Q8—81no—
205C3×D4—121no—
206C3×D4—121no—
207C3×D4—121no—
208C3×D4—121no—
209C3×D4—241no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4#23511yescharacteristic
217C2×C3×C4D3×D411yescharacteristic
218C2×C3×C4D3×D411yescharacteristic
219C2×C3×C4D3×D411yescharacteristic
220C2×C3×C4#23711yescharacteristic
221C2×C3×C4C2×#6711yescharacteristic
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—121no—
227C2×Dic3—121no—
228D12—41no—
229D12—41no—
230D12—41no—
231D12—41no—
232D12—81no—
233Dic6—62no—
234Dic6—61no—
235Dic6—61no—
236Dic6—61no—
237Dic6—61no—
238Dic6—121no—
239C3×C8—32no—
240C3×C8—61no—
241C3×C8—121no—
242C3×C8—121no—
243C3⋊C8—21no—
244C3⋊C8—41no—
245C3⋊C8—41no—
246C3⋊C8—122no—
247C3⋊C8—121no—
248C3⋊C8—121no—
249C3⋊C8#21412yescomplemented
250C2×C32#46011yescharacteristic
251C2×C32#46811yescharacteristic
252C2×C32#47111yescharacteristic
253C3×D3—161no—
254C2×Q8—21no—
255C2×Q8—91no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259SD16—122no—
260SD16—122no—
261SD16—362no—
262C2×C8—31no—
263C2×C8—31no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—181no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C42—181no—
272C42C2×D3211yescharacteristic
273C22×C3—21no—
274C22×C3—241no—
275C22×C3—241no—
276C22×C3—241no—
277C22×C3#88411yescharacteristic
278C22×C3#93811yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—82no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4#91111yescharacteristic
293C3×C4D3×SD1611yescharacteristic
294C3×C4#91411yescharacteristic
295C3×C4#91411yescharacteristic
296C3×C4#91511yescharacteristic
297C3×C4#91811yescharacteristic
298C3×C4#93211yescharacteristic
299C3×C4C2×#21411yescharacteristic
300Dic3—122no—
301Dic3—121no—
302Dic3—121no—
303Dic3—121no—
304Dic3—121no—
305Dic3—241no—
306C32#168611yescharacteristic, complemented, Sylow
307C23—121no—
308Q8—41no—
309Q8—41no—
310Q8—181no—
311Q8—181no—
312D4—121no—
313D4—121no—
314D4—121no—
315D4—121no—
316D4—241no—
317C2×C4—41no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4#380411yescharacteristic
321C2×C4D3×#6711yescharacteristic
322C2×C4#381111yescharacteristic
323C8—32no—
324C8—61no—
325C8—121no—
326C8—121no—
327C8—361no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—481no—
332C2×C3#475411yescharacteristic
333C2×C3#476211yescharacteristic
334C2×C3#481211yescharacteristic
335C2×C3#507511yescharacteristic
336C2×C3#517411yescharacteristic
337C2×C3#517711yescharacteristic
338D3—161no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#6395811yescharacteristic
343C4—21no—
344C4—81no—
345C4—361no—
346C4—361no—
347C4D3×#21411yescharacteristic
348C4#6393211yescharacteristic
349C4#6394111yescharacteristic
350C4#6395111yescharacteristic
351C3—21no—
352C3#6847211yescharacteristic, complemented
353C3#7127311yescharacteristic, complemented
354C2—481no—
355C2#1059143911yescharacteristic
356C2#1059154211yescharacteristic
357C2#1059159011yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4⋊Q8D32
hypocenterC32#1686
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. #10592793C2semidirect
4. #10590296C2semidirect
5. C3×#4547C2semidirect
6. #10589099C2semidirect
35. #4547D3semidirect
115. C3×C3⋊C8SD16semidirect
249. C3⋊C8#214semidirect
306. C32#1686semidirect
352. C3#68472semidirect
353. C3#71273semidirect

Maximal subgroups

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

subgroupindex
#105927932
C3×#50852
#105903772
#105902962
C3×#45472
#105890992
#105890902
#712733
#684723