Finite Group Databasebeta

← #11781674

Subgroups — #11781674

Order 1152.

383 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 88 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#10592723C211yesmaximal, characteristic, complemented
2C3×#5027C211yesmaximal, characteristic
3C3×#5020C211yesmaximal, characteristic, complemented
4#10590375C211yesmaximal, characteristic
5#10589100C211yesmaximal, characteristic, complemented
6#10589092C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71120—31nomaximal
9#71024—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C2×C3×#215—21no—
12C3×#874C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3715—21no—
15C2×#3715—21no—
16C4⋊(C3⋊(C3⋊C8))—21no—
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18#63630—21no—
19C4×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×#835—21no—
21C3×#809—21no—
22C3×#808—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25C3×#475—31no—
26C3×#460—31no—
27#5085—31no—
28#5076—31no—
29#5027D311yescharacteristic
30#5024—31no—
31#5024—31no—
32#5024—61no—
33#5020D311yescharacteristic, complemented
34#4547—31no—
35#4544—31no—
36#4486—31no—
37#4485—31no—
38#4479—31no—
39#4479—31no—
40C2×C32×Q8D411yescharacteristic
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×Dic6—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C42—21no—
48C32×C42C2311yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C2×C3⋊(C3⋊C8)D411yescharacteristic
52C3×#215—42no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57#3715—42no—
58#3715—42no—
59#1686—91noSylow
60C2×C3×SD16—61no—
61C3×C4⋊Q8—31no—
62C3×#127—31no—
63C3×C4×Q8—21no—
64C3×C4×Q8C2×D311yescharacteristic
65C3×C4×Q8C2×D311yescharacteristic
66C2×#215—21no—
67C2×#215—61no—
68C2×#215—61no—
69C2×#214—61no—
70C2×#214—61no—
71#874C2×D311yescharacteristic
72#869C2×D311yescharacteristic
73C3×#106—61no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8—31no—
76C3×#103—61no—
77C3×#102—61no—
78#835—21no—
79#809—21no—
80#808—21no—
81C4⋊(C3⋊C8)—31no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—61no—
84C4⋊(C3⋊C8)—61no—
85C4⋊(C3⋊C8)—61no—
86C4⋊(C3⋊C8)—62no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89#803—61no—
90#803—61no—
91#803—62no—
92C4×C3⋊C8—31no—
93C4×C3⋊C8—31no—
94C4×C3⋊C8—61no—
95C22×C3×D3—41no—
96C32×Q8—21no—
97C32×Q8—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4#4111yescharacteristic
103C2×C3×Dic3—41no—
104C3×D12—41no—
105C3×D12—41no—
106C3×D12—41no—
107C3×D12—41no—
108C3×D12—81no—
109C3×Dic6—41no—
110C3×Dic6—41no—
111C3⋊(C3⋊C8)—21no—
112C3⋊(C3⋊C8)—41no—
113C3⋊(C3⋊C8)SD1612yescomplemented
114C3×C3⋊C8—41no—
115C3×C3⋊C8—41no—
116#492—91no—
117#475—31no—
118#460—31no—
119#446—91no—
120#335—91no—
121#332—91no—
122#331—91no—
123C2×C3×Q8—21no—
124C2×C3×Q8—61no—
125C2×C3×Q8#6711yescharacteristic
126C2×C3×Q8#6711yescharacteristic
127C2×C3×D4—61no—
128C2×C3×D4—61no—
129C2×C3×D4—61no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×Dic6—21no—
134C3×SD16—122no—
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—61no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4#6711yescharacteristic
144C3×C4⋊C4#6711yescharacteristic
145C3×C42—22no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42C22×D311yescharacteristic
150C3×C42C22×D311yescharacteristic
151#215—42no—
152#215—122no—
153#215—122no—
154#214—122no—
155#214—122no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8—62no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—61no—
169C22×C32#12111yescharacteristic
170C2×C3×D3—81no—
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4C2×SD1611yescharacteristic
177C32×C4AGL(1,ℤ/8)11yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C32×C4#13711yescharacteristic
180C3×Dic3—81no—
181C2×SD16—61no—
182C2×SD16—181no—
183C2×SD16—181no—
184C4⋊Q8—31no—
185#127—31no—
186C4×Q8D3211yescharacteristic
187#106—61no—
188C4⋊C8—31no—
189C4⋊C8—31no—
190C4⋊C8—91no—
191C4⋊C8—181no—
192#103—61no—
193#102—61no—
194#97—181no—
195C4×C8—91no—
196C23×C3—121no—
197C22×D3—41no—
198C3×Q8—21no—
199C3×Q8—21no—
200C3×Q8—21no—
201C3×Q8—21no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—121no—
205C3×Q8—121no—
206C3×D4—121no—
207C3×D4—121no—
208C3×D4—121no—
209C3×D4—121no—
210C3×D4—241no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—121no—
221C2×C3×C4D3×D411yescharacteristic
222C2×C3×C4D3×D411yescharacteristic
223C2×C3×C4#23711yescharacteristic
224C2×C3×C4#23711yescharacteristic
225C2×C3×C4C2×#6711yescharacteristic
226C2×C3×C4C2×#6711yescharacteristic
227C2×Dic3—41no—
228D12—41no—
229D12—41no—
230D12—41no—
231D12—41no—
232D12—81no—
233Dic6—41no—
234Dic6—41no—
235C3×C8—121no—
236C3×C8—121no—
237C3⋊C8—32no—
238C3⋊C8—32no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—62no—
242C3⋊C8—61no—
243C3⋊C8—61no—
244C3⋊C8—121no—
245C3⋊C8—122no—
246C3⋊C8—121no—
247C3⋊C8—121no—
248C2×C32#46011yescharacteristic
249C2×C32#46811yescharacteristic
250C2×C32#47111yescharacteristic
251C3×D3—161no—
252C2×Q8—61no—
253C2×Q8#62511yescharacteristic
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257SD16—122no—
258SD16—362no—
259SD16—362no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—91no—
263C2×C8—91no—
264C2×C8—181no—
265C4⋊C4—21no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4#62511yescharacteristic
269C42—21no—
270C42C2×D3211yescharacteristic
271C22×C3—21no—
272C22×C3—241no—
273C22×C3—241no—
274C22×C3—241no—
275C22×C3#93911yescharacteristic
276C22×C3#93911yescharacteristic
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—81no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—42no—
288C3×C4—42no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—241no—
294C3×C4D3×SD1611yescharacteristic
295C3×C4D3×SD1611yescharacteristic
296C3×C4#91411yescharacteristic
297C3×C4#91411yescharacteristic
298C3×C4#93211yescharacteristic
299C3×C4#94211yescharacteristic
300C3×C4#94911yescharacteristic
301C3×C4#95111yescharacteristic
302Dic3—81no—
303C32#168611yescharacteristic, complemented, Sylow
304C23—121no—
305Q8—21no—
306Q8—21no—
307Q8—121no—
308Q8—121no—
309D4—121no—
310D4—121no—
311D4—121no—
312D4—121no—
313D4—241no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—121no—
317C2×C4#380511yescharacteristic
318C2×C4C2×#62511yescharacteristic
319C2×C4#381111yescharacteristic
320C8—92no—
321C8—121no—
322C8—121no—
323C8—181no—
324C8—361no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—481no—
329C2×C3#504211yescharacteristic
330C2×C3#504911yescharacteristic
331C2×C3#516311yescharacteristic
332C2×C3#516311yescharacteristic
333C2×C3#520011yescharacteristic
334C2×C3#520311yescharacteristic
335D3—161no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#6398011yescharacteristic
340C4—21no—
341C4—41no—
342C4—41no—
343C4—241no—
344C4#6382311yescharacteristic
345C4#6382711yescharacteristic
346C4#6394011yescharacteristic
347C4#6394111yescharacteristic
348C3—21no—
349C3#7102411yescharacteristic, complemented
350C3#7112011yescharacteristic, complemented
351C2—481no—
352C2#1059118011yescharacteristic
353C2#1059162211yescharacteristic
354C2#1059168111yescharacteristic
subgroupG/H
centerC22#63980
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. #10592723C2semidirect
3. C3×#5020C2semidirect
5. #10589100C2semidirect
6. #10589092C2semidirect
33. #5020D3semidirect
113. C3⋊(C3⋊C8)SD16semidirect
303. C32#1686semidirect
349. C3#71024semidirect
350. C3#71120semidirect

Maximal subgroups

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

subgroupindex
#105927232
C3×#50272
C3×#50202
#105903752
#105891002
#105890922
#105890902
#711203
#710243