Finite Group Databasebeta

← #11781681

Subgroups — #11781681

Order 1152.

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

normal subgroups only

This group has 368 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
1C3×#4848C211yesmaximal, characteristic
2#10590369C211yesmaximal, characteristic
3#10590301C211yesmaximal, characteristic, complemented
4#10590258C211yesmaximal, characteristic, complemented
5#10589352C211yesmaximal, characteristic, complemented
6C3×#4479C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71024—31nomaximal
9#68592—31nomaximal
10#64078C2211yescharacteristic
11C2×C3×#204—21no—
12C3×#874C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14C2×#3716—21no—
15C2×#3716—21no—
16C32×C4⋊C8C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#821—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63573—21no—
22Dic3⋊(C3⋊C8)—21no—
23#63566—21no—
24#63554—21no—
25C3×#460—31no—
26#5076—31no—
27#5024—31no—
28#5020—31no—
29#4851—31no—
30#4848D311yescharacteristic
31#4693—31no—
32#4679—31no—
33C3×#335—31no—
34#4544—31no—
35#4486—31no—
36#4486—31no—
37#4486—61no—
38#4485—31no—
39#4479D311yescharacteristic, complemented
40C2×#634—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×Dic6D411yescharacteristic
45C2×C32×C8—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊C8)—21no—
50C3×C4⋊Dic3D411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×#204—42no—
57#3716—42no—
58#3716—42no—
59#1686—91noSylow
60C2×C3×SD16—61no—
61C3×#127—31no—
62C3×C4×Q8—31no—
63C2×#214—61no—
64C2×#214—61no—
65C2×#204—21no—
66C2×#204—61no—
67C2×#204—61no—
68#874C2×D311yescharacteristic
69#869—31no—
70#869—31no—
71#869—61no—
72C4×Dic6C2×D311yescharacteristic
73C3×C4⋊C8—21no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8C2×D311yescharacteristic
76C3×C4⋊C8C2×D311yescharacteristic
77C3×#102—61no—
78C3×C4×C8—31no—
79#835—61no—
80#821—21no—
81#817—61no—
82#816—61no—
83#815—61no—
84Dic3⋊C8—61no—
85#808—61no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91#803—61no—
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C3⋊Dic3—41no—
98#634—41no—
99#634—41no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C3×D12—41no—
103C3×D12—41no—
104C3×D12—41no—
105C3×D12—41no—
106C3×D12—81no—
107C3×Dic6—21no—
108C3×Dic6—21no—
109C32×C8—41no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8SD1612yescomplemented
114#492—91no—
115#475—91no—
116#460—31no—
117#446—91no—
118#335—31no—
119#332—91no—
120#331—91no—
121C2×C3×Q8—31no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×Dic6—61no—
129C2×Dic6—61no—
130C2×Dic6—121no—
131C2×Dic6#6711yescharacteristic
132C3×SD16—122no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—41no—
138C2×C3×C8—61no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—61no—
141C3×C42—21no—
142C3×C42—61no—
143C3×C42C22×D311yescharacteristic
144C3×C42C22×D311yescharacteristic
145#214—122no—
146#214—122no—
147C4⋊Dic3—62no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—121no—
153C4⋊Dic3#6711yescharacteristic
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—121no—
160C2×C3⋊C8D1211yescharacteristic
161C2×C3⋊C8D1211yescharacteristic
162#204—42no—
163#204—122no—
164#204—122no—
165C22×C32#12111yescharacteristic
166C2×C3×D3—81no—
167C2×C3×D3—81no—
168C2×C3×D3—81no—
169C32×C4—21no—
170C32×C4C2×SD1611yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C32×C4AGL(1,ℤ/8)11yescharacteristic
173C32×C4#13711yescharacteristic
174C3⋊Dic3—81no—
175C3×Dic3—41no—
176C3×Dic3—41no—
177C2×SD16—61no—
178C2×SD16—181no—
179C2×SD16—181no—
180C4⋊Q8—91no—
181#127—31no—
182C4×Q8—31no—
183#106—181no—
184C4⋊C8—31no—
185C4⋊C8—91no—
186C4⋊C8—181no—
187C4⋊C8D3211yescharacteristic
188#103—181no—
189#102—61no—
190#97—181no—
191C4×C8—31no—
192C23×C3—121no—
193C22×D3—41no—
194C3×Q8—61no—
195C3×Q8—61no—
196C3×D4—121no—
197C3×D4—121no—
198C3×D4—121no—
199C3×D4—121no—
200C3×D4—241no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4C2×D1211yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4#23711yescharacteristic
210C2×C3×C4#23911yescharacteristic
211C2×C3×C4C2×#6711yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—122no—
215C2×Dic3—121no—
216C2×Dic3—121no—
217D12—41no—
218D12—41no—
219D12—41no—
220D12—41no—
221D12—81no—
222Dic6—21no—
223Dic6—21no—
224Dic6—121no—
225Dic6—121no—
226Dic6—121no—
227Dic6—121no—
228Dic6—241no—
229Dic6—241no—
230C3×C8—32no—
231C3×C8—41no—
232C3×C8—41no—
233C3×C8—61no—
234C3×C8—81no—
235C3×C8—121no—
236C3⋊C8—21no—
237C3⋊C8—41no—
238C3⋊C8—121no—
239C3⋊C8—121no—
240C3⋊C8—241no—
241C3⋊C8#20412yescomplemented
242C2×C32#46011yescharacteristic
243C2×C32#46811yescharacteristic
244C2×C32#47111yescharacteristic
245C3×D3—161no—
246C2×Q8—31no—
247C2×Q8—181no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251SD16—122no—
252SD16—362no—
253SD16—362no—
254C2×C8—21no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—61no—
258C2×C8—181no—
259C4⋊C4—31no—
260C4⋊C4—61no—
261C4⋊C4—181no—
262C4⋊C4—181no—
263C42—61no—
264C42C2×D3211yescharacteristic
265C22×C3—21no—
266C22×C3—241no—
267C22×C3—241no—
268C22×C3—241no—
269C22×C3#89511yescharacteristic
270C22×C3#93911yescharacteristic
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—121no—
282C3×C4—121no—
283C3×C4C2×#20411yescharacteristic
284C3×C4#90811yescharacteristic
285C3×C4#91111yescharacteristic
286C3×C4D3×SD1611yescharacteristic
287C3×C4#91411yescharacteristic
288C3×C4#91811yescharacteristic
289C3×C4#93211yescharacteristic
290C3×C4#95111yescharacteristic
291Dic3—41no—
292Dic3—41no—
293Dic3—242no—
294Dic3—241no—
295Dic3—241no—
296C32#168611yescharacteristic, complemented, Sylow
297C23—121no—
298Q8—61no—
299Q8—61no—
300Q8—361no—
301Q8—361no—
302D4—121no—
303D4—121no—
304D4—121no—
305D4—121no—
306D4—241no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—361no—
310C2×C4#379611yescharacteristic
311C2×C4D3×D1211yescharacteristic
312C2×C4D3×#6711yescharacteristic
313C8—32no—
314C8—41no—
315C8—61no—
316C8—121no—
317C8—361no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—481no—
322C2×C3#484811yescharacteristic
323C2×C3#488611yescharacteristic
324C2×C3#489811yescharacteristic
325C2×C3#504211yescharacteristic
326C2×C3#516311yescharacteristic
327C2×C3#520011yescharacteristic
328D3—161no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22#6388611yescharacteristic
333C4—21no—
334C4—121no—
335C4—121no—
336C4—721no—
337C4D3×#20411yescharacteristic
338C4#6379411yescharacteristic
339C4#6393211yescharacteristic
340C4#6394311yescharacteristic
341C3—21no—
342C3#6859211yescharacteristic, complemented
343C3#7102411yescharacteristic, complemented
344C2—481no—
345C2#1059058811yescharacteristic
346C2#1059079911yescharacteristic
347C2#1059098611yescharacteristic
subgroupG/H
centerC22#63886
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4⋊C8D32
hypocenterC32#1686
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #10590301C2semidirect
4. #10590258C2semidirect
5. #10589352C2semidirect
6. C3×#4479C2semidirect
39. #4479D3semidirect
113. C3×C3⋊C8SD16semidirect
241. C3⋊C8#204semidirect
296. C32#1686semidirect
342. C3#68592semidirect
343. C3#71024semidirect

Maximal subgroups

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

subgroupindex
C3×#48482
#105903692
#105903012
#105902582
#105893522
C3×#44792
#105890902
#710243
#685923