Finite Group Databasebeta

← #11781031

Subgroups — #11781031

Order 1152.

399 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 96 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
1D4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C3×#5019C211yesmaximal, characteristic
3C3×#5008C211yesmaximal, characteristic
4#10590309C211yesmaximal, characteristic, complemented
5#10589098C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71132—31nomaximal
9#71036—31nomaximal
10C32×C4×D4C2211yescharacteristic
11C22×C3⋊(C3⋊C8)—21no—
12C2×C3×#214—21no—
13C3×#869C2211yescharacteristic
14C3×#869C2211yescharacteristic
15#63657—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#832—21no—
19C3×#810—21no—
20C3×#808—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23#63574—21no—
24#63574—21no—
25C3×#476—31no—
26C3×#461—31no—
27#5080—31no—
28#5080—31no—
29#5019D311yescharacteristic
30D4×C3⋊C8—31no—
31D4×C3⋊C8—31no—
32D4×C3⋊C8—61no—
33#5008D311yescharacteristic
34#4547—31no—
35#4546—31no—
36#4482—31no—
37#4481—31no—
38#4480—31no—
39#4480—31no—
40C2×C32×D4D411yescharacteristic
41C22×C32×C4—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4D411yescharacteristic
45C32×C22⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C2×C3⋊(C3⋊C8)—42no—
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C3×#214—42no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—21no—
56C2×C3×C3⋊C8—21no—
57C2×C3×C3⋊C8—21no—
58#1671—91noSylow
59C2×C3×SD16—61no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8—31no—
62C3×C4×D4—21no—
63C3×C4×D4C2×D311yescharacteristic
64C3×C4×D4C2×D311yescharacteristic
65C2×#214—21no—
66C22×C3⋊C8—61no—
67C22×C3⋊C8—61no—
68C22×C3⋊C8—62no—
69#869C2×D311yescharacteristic
70#869C2×D311yescharacteristic
71C3×#106—61no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×#103—61no—
75C3×#102—61no—
76#832—21no—
77#830—61no—
78#830—61no—
79#830—62no—
80#810—21no—
81#808—21no—
82#808—61no—
83#808—61no—
84#808—61no—
85#808—61no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8—31no—
92C4×C3⋊C8—31no—
93C4×C3⋊C8—61no—
94C23×C32—21no—
95C32×D4—21no—
96C32×D4SD1612yes—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4#4111yescharacteristic
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C3×Dic6—81no—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)—41no—
110C3⋊(C3⋊C8)SD1612yescomplemented
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113#498—91no—
114#476—31no—
115#461—31no—
116C8×D4—91no—
117#334—91no—
118#333—91no—
119#332—91no—
120C2×C3×Q8—61no—
121C2×C3×Q8—61no—
122C2×C3×D4—21no—
123C2×C3×D4#6711yescharacteristic
124C2×C3×D4#6711yescharacteristic
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×SD16—122no—
131C2×C3×C8—61no—
132C2×C3×C8—61no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4#6711yescharacteristic
139C3×C4⋊C4#6711yescharacteristic
140C3×C22⋊C4—22no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C42—21no—
144C3×C42C22×D311yescharacteristic
145C3×C42C22×D311yescharacteristic
146#214—42no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—122no—
163C2×C3⋊C8—122no—
164C2×C3⋊C8—124no—
165C22×C32—22no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32#12111yescharacteristic
169C32×C4—21no—
170C32×C4—41no—
171C32×C4C2×SD1611yescharacteristic
172C32×C4C2×SD1611yescharacteristic
173C32×C4#13511yescharacteristic
174C32×C4#13711yescharacteristic
175C3×Dic3—81no—
176C3×Dic3—81no—
177C2×SD16—61no—
178C22×C8—181no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181C4×D4D3211yescharacteristic
182#106—61no—
183#106—181no—
184#106—181no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—91no—
188#103—61no—
189#102—61no—
190C22⋊C8—181no—
191C4×C8—91no—
192C23×C3—22no—
193C23×C3—21no—
194C23×C3—21no—
195C3×Q8—121no—
196C3×Q8—121no—
197C3×Q8—241no—
198C3×D4—22no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—22no—
202C3×D4#21412yes—
203C3×D4#21512yes—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—121no—
217C2×C3×C4—121no—
218C2×C3×C4D3×D411yescharacteristic
219C2×C3×C4D3×D411yescharacteristic
220C2×C3×C4#23711yescharacteristic
221C2×C3×C4#23711yescharacteristic
222C2×C3×C4C2×#6711yescharacteristic
223C2×C3×C4C2×#6711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—41no—
226Dic6—41no—
227Dic6—41no—
228Dic6—81no—
229C3×C8—121no—
230C3×C8—121no—
231C3⋊C8—32no—
232C3⋊C8—32no—
233C3⋊C8—41no—
234C3⋊C8—41no—
235C3⋊C8—62no—
236C3⋊C8—61no—
237C3⋊C8—61no—
238C3⋊C8—121no—
239C3⋊C8—122no—
240C3⋊C8—121no—
241C3⋊C8—121no—
242C2×C32—42no—
243C2×C32#46111yescharacteristic
244C2×C32#46511yescharacteristic
245C2×C32#47111yescharacteristic
246C2×Q8—61no—
247C2×Q8—61no—
248C2×D4#62511yescharacteristic
249C22×C4—21no—
250SD16—122no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—91no—
254C2×C8—91no—
255C2×C8—181no—
256C2×C8—362no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4#62511yescharacteristic
262C22⋊C4—21no—
263C42C2×D3211yescharacteristic
264C22×C3—24no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3#93911yescharacteristic
275C22×C3#93911yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—42no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—241no—
287C3×C4—241no—
288C3×C4D3×SD1611yescharacteristic
289C3×C4D3×SD1611yescharacteristic
290C3×C4#91511yescharacteristic
291C3×C4#91511yescharacteristic
292C3×C4C2×#21411yescharacteristic
293C3×C4C2×#21511yescharacteristic
294C3×C4#95011yescharacteristic
295C3×C4#95011yescharacteristic
296Dic3—81no—
297Dic3—81no—
298C32#167111yescharacteristic, complemented, Sylow
299C23—21no—
300Q8—121no—
301Q8—121no—
302Q8—241no—
303D4—21no—
304D4#371512yes—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4#380511yescharacteristic
311C2×C4C2×#62511yescharacteristic
312C2×C4#381111yescharacteristic
313C8—92no—
314C8—121no—
315C8—121no—
316C8—181no—
317C8—361no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—44no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3#504111yescharacteristic
325C2×C3#504811yescharacteristic
326C2×C3#516211yescharacteristic
327C2×C3#516211yescharacteristic
328C2×C3#520011yescharacteristic
329C2×C3#520311yescharacteristic
330C22—22no—
331C22—41no—
332C22—41no—
333C22#6398011yescharacteristic
334C4—21no—
335C4—41no—
336C4—241no—
337C4—241no—
338C4#6382211yescharacteristic
339C4C2×#371511yescharacteristic
340C4#6393011yescharacteristic
341C4#6393111yescharacteristic
342C3—21no—
343C3#7103611yescharacteristic, complemented
344C3#7113211yescharacteristic, complemented
345C2—42no—
346C2#1059117911yescharacteristic
347C2#1059149711yescharacteristic
348C2#1059168111yescharacteristic
subgroupG/H
centerC22#63980
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32#121
prime radicalC2×C32×D4D4
hypercenterC4×D4D32
hypocenterC32#1671
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
4. #10590309C2semidirect
5. #10589098C2semidirect
6. #10589095C2semidirect
7. #10589093C2semidirect
110. C3⋊(C3⋊C8)SD16semidirect
298. C32#1671semidirect
343. C3#71036semidirect
344. C3#71132semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊C8)2
C3×#50192
C3×#50082
#105903092
#105890982
#105890952
#105890932
#711323
#710363