Finite Group Databasebeta

← #11781305

Subgroups — #11781305

Order 1152.

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

normal subgroups only

This group has 379 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#10592787C211yesmaximal, characteristic
2C3×#5091C211yesmaximal, characteristic, complemented
3#10590367C211yesmaximal, characteristic
4#10590250C211yesmaximal, characteristic, complemented
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589098C211yesmaximal, characteristic, complemented
7#10589091C211yesmaximal, characteristic
8#71150—31nomaximal
9#68510—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64083C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C2×#3717—21no—
14#63636—21no—
15#63636—21no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#810—21no—
18C3×#807—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22Dic3⋊(C3⋊C8)—21no—
23#63564—21no—
24Dic3×C3⋊C8—21no—
25C3×#477—31no—
26#5091D311yescharacteristic, complemented
27#5088—31no—
28#5088—31no—
29#5088—61no—
30#5027—31no—
31Q8×C3⋊C8—31no—
32#4848—31no—
33#4830—31no—
34C3×#333—31no—
35#4546D311yescharacteristic, complemented
36#4485—31no—
37#4485—31no—
38#4480—31no—
39#4480—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C3×Dic6D411yescharacteristic
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C42C2311yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C3×C4⋊Dic3D411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56#3717—42no—
57#1678—91noSylow
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4×Q8—31no—
62C2×#215—61no—
63C2×#204—61no—
64#874—31no—
65#874—31no—
66#874—61no—
67C4×Dic6C2×D311yescharacteristic
68C3×#107—61no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—31no—
72C3×#103—61no—
73#821—61no—
74Dic3⋊C8—61no—
75C8×Dic3—61no—
76#810—21no—
77#809—61no—
78#809—61no—
79#809—61no—
80#809—61no—
81#809—61no—
82#809—62no—
83#809—62no—
84#807—21no—
85C4⋊(C3⋊C8)—61no—
86C4⋊(C3⋊C8)C2×D311yescharacteristic
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4×C3⋊C8—31no—
90C4×C3⋊C8—31no—
91C4×C3⋊C8—61no—
92C4×C3⋊C8—61no—
93C22×C3⋊D3—41no—
94C32×Q8—81no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C4⋊(C3⋊D3)—41no—
100C4⋊(C3⋊D3)—41no—
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)—41no—
103C4⋊(C3⋊D3)—81no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C3×Dic6—21no—
107C3×Dic6SD1612yes—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113#486—91no—
114#477—31no—
115#460—91no—
116C8×Q8—91no—
117#334—91no—
118#333—31no—
119#331—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—31no—
123C2×C3×Q8—41no—
124C2×D12—62no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—61no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—61no—
131C2×D12—121no—
132C2×D12—121no—
133C2×Dic6#6711yescharacteristic
134C2×C3×C8—61no—
135C2×C3×C8—61no—
136C2×C3×C8—61no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—61no—
145C3×C42—21no—
146C3×C42—61no—
147C3×C42C22×D311yescharacteristic
148C3×C42C22×D311yescharacteristic
149#215—122no—
150C4⋊Dic3#6711yescharacteristic
151Dic3⋊C4—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—61no—
162#204—122no—
163C22×C32#12111yescharacteristic
164C2×C3⋊D3—81no—
165C2×C3⋊D3—81no—
166C2×C3⋊D3—81no—
167C32×C4—21no—
168C32×C4—81no—
169C32×C4C2×SD1611yescharacteristic
170C32×C4#13511yescharacteristic
171C32×C4#13511yescharacteristic
172C32×C4#13711yescharacteristic
173C3×Dic3—22no—
174C3×Dic3—41no—
175C2×SD16—181no—
176C4⋊Q8D3211yescharacteristic
177#127—91no—
178C4×Q8—31no—
179#107—61no—
180C4⋊C8—31no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—181no—
184#103—61no—
185#102—181no—
186#102—181no—
187#102—181no—
188C4×C8—91no—
189C4×C8—181no—
190C22×D3—122no—
191C22×D3—121no—
192C22×D3—121no—
193C3×Q8—32no—
194C3×Q8—61no—
195C3×Q8—81no—
196C3×Q8—81no—
197C3×Q8—82no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4#23511yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4D3×D411yescharacteristic
210C2×C3×C4#23711yescharacteristic
211C2×C3×C4C2×#6711yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—21no—
214D12—122no—
215D12—122no—
216D12—121no—
217D12—121no—
218D12—121no—
219D12—121no—
220D12—121no—
221D12—121no—
222D12—121no—
223D12—121no—
224D12—241no—
225D12—241no—
226D12—241no—
227D12—241no—
228D12—242no—
229Dic6—21no—
230Dic6#21512yes—
231C3×C8—121no—
232C3×C8—121no—
233C3×C8—121no—
234C3⋊C8—41no—
235C3⋊C8—41no—
236C3⋊C8—41no—
237C3⋊C8—61no—
238C3⋊C8—61no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—121no—
242C3⋊C8—121no—
243C2×C32#46411yescharacteristic
244C2×C32#46511yescharacteristic
245C2×C32#47211yescharacteristic
246C3⋊D3—161no—
247C2×Q8—21no—
248C2×Q8—31no—
249C2×D4—181no—
250C2×D4—181no—
251C2×D4—181no—
252SD16—362no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—91no—
257C2×C8—91no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—31no—
261C4⋊C4—61no—
262C42—61no—
263C42C2×D3211yescharacteristic
264C22×C3—21no—
265C22×C3#88411yescharacteristic
266C22×C3#93911yescharacteristic
267C2×D3—242no—
268C2×D3—242no—
269C2×D3—242no—
270C2×D3—241no—
271C2×D3—241no—
272C2×D3—241no—
273C2×D3—241no—
274C2×D3—241no—
275C2×D3—241no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—62no—
284C3×C4—82no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—121no—
288C3×C4D3×SD1611yescharacteristic
289C3×C4#91611yescharacteristic
290C3×C4#91811yescharacteristic
291C3×C4#91811yescharacteristic
292C3×C4#91911yescharacteristic
293C3×C4#91911yescharacteristic
294C3×C4C2×#21511yescharacteristic
295C3×C4#95011yescharacteristic
296Dic3—22no—
297Dic3—41no—
298C32#167811yescharacteristic, complemented, Sylow
299C23—361no—
300Q8—32no—
301Q8—61no—
302Q8—81no—
303D4—361no—
304D4—361no—
305D4—361no—
306D4—361no—
307D4—721no—
308C2×C4—41no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4#380411yescharacteristic
312C2×C4D3×#6711yescharacteristic
313C2×C4#381111yescharacteristic
314C8—121no—
315C8—121no—
316C8—121no—
317C8—181no—
318C8—181no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#474911yescharacteristic
323C2×C3#479811yescharacteristic
324C2×C3#480711yescharacteristic
325C2×C3#504811yescharacteristic
326C2×C3#518511yescharacteristic
327C2×C3#519911yescharacteristic
328D3—482no—
329D3—481no—
330D3—481no—
331C22—721no—
332C22—721no—
333C22—721no—
334C22#6397111yescharacteristic
335C4—21no—
336C4—62no—
337C4—81no—
338C4—121no—
339C4D3×#21511yescharacteristic
340C4#6394511yescharacteristic
341C4#6394911yescharacteristic
342C4#6395011yescharacteristic
343C3—21no—
344C3#6851011yescharacteristic, complemented
345C3#7115011yescharacteristic, complemented
346C2—1441no—
347C2#1059150811yescharacteristic
348C2#1059157911yescharacteristic
349C2#1059158511yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radical#64083C22
hypercenterC4⋊Q8D32
hypocenterC32#1678
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
2. C3×#5091C2semidirect
4. #10590250C2semidirect
5. C3×#4546C2semidirect
6. #10589098C2semidirect
26. #5091D3semidirect
35. #4546D3semidirect
298. C32#1678semidirect
344. C3#68510semidirect
345. C3#71150semidirect

Maximal subgroups

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

subgroupindex
#105927872
C3×#50912
#105903672
#105902502
C3×#45462
#105890982
#105890912
#711503
#685103