Finite Group Databasebeta

← #11781678

Subgroups — #11781678

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×#5094C211yesmaximal, characteristic, complemented
2C3×#5085C211yesmaximal, characteristic
3#10590364C211yesmaximal, characteristic
4#10590258C211yesmaximal, characteristic, complemented
5#10589413C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71273—31nomaximal
9#71120—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#215—21no—
12C2×C3×#215—21no—
13C3×#874C2211yescharacteristic
14C3×C4×Dic6C2211yescharacteristic
15C2×#3715—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#810—21no—
19C3×#808—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22Dic3⋊(C3⋊C8)—21no—
23#63562—21no—
24#63554—21no—
25C3×#492—31no—
26C3×#475—31no—
27#5094D311yescharacteristic, complemented
28#5085D311yescharacteristic
29#5075—31no—
30#5027—31no—
31#5024—31no—
32#4679—31no—
33#4547—31no—
34#4547—31no—
35#4547—61no—
36#4544—31no—
37#4485—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×Dic6D411yescharacteristic
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C42C2311yescharacteristic
48C2×C3⋊(C3⋊C8)—21no—
49C2×C3⋊(C3⋊C8)—21no—
50C3×#215—42no—
51C3×#215—42no—
52C3×C4⋊Dic3D411yescharacteristic
53C3×Dic3⋊C4—21no—
54C3×C4×Dic3—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#3715—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—
66C3×C4×Q8—31no—
67C2×#215—21no—
68C2×#215—21no—
69C2×#215—61no—
70C2×#214—61no—
71#874C2×D311yescharacteristic
72C4×Dic6C2×D311yescharacteristic
73C3×#106—61no—
74C3×C4⋊C8—31no—
75C3×#103—61no—
76C3×C4×C8—31no—
77#832—61no—
78#815—61no—
79Dic3⋊C8—61no—
80#810—21no—
81#809—61no—
82#808—21no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91#803—61no—
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—41no—
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
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—
109C3⋊(C3⋊C8)—41no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8SD1612yescomplemented
114#492—31no—
115#475—31no—
116#460—91no—
117#446—91no—
118#335—91no—
119#332—91no—
120#331—91no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—31no—
124C2×C3×Q8—41no—
125C2×C3×D4—61no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×Dic6#6711yescharacteristic
132C3×SD16—122no—
133C3×SD16—122no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
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—42no—
150#215—42no—
151#215—122no—
152#214—122no—
153C4⋊Dic3#6711yescharacteristic
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—121no—
162C2×C3⋊C8—121no—
163C2×C3⋊C8#6711yescharacteristic
164C2×C3⋊C8#6711yescharacteristic
165C22×C32#12111yescharacteristic
166C2×C3×D3—81no—
167C2×C3×D3—81no—
168C2×C3×D3—81no—
169C32×C4—21no—
170C32×C4—81no—
171C32×C4C2×SD1611yescharacteristic
172C32×C4AGL(1,ℤ/8)11yescharacteristic
173C32×C4AGL(1,ℤ/8)11yescharacteristic
174C32×C4#13711yescharacteristic
175C3×Dic3—41no—
176C3×Dic3—41no—
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—181no—
180C4⋊Q8D3211yescharacteristic
181#127—31no—
182C4×Q8—31no—
183#106—61no—
184C4⋊C8—31no—
185C4⋊C8—91no—
186C4⋊C8—91no—
187C4⋊C8—181no—
188#103—61no—
189#102—181no—
190#97—181no—
191C4×C8—31no—
192C23×C3—121no—
193C22×D3—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—41no—
198C3×Q8—61no—
199C3×Q8—61no—
200C3×Q8—81no—
201C3×Q8—81no—
202C3×D4—121no—
203C3×D4—121no—
204C3×D4—121no—
205C3×D4—121no—
206C3×D4—241no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4#23711yescharacteristic
218C2×C3×C4#23711yescharacteristic
219C2×C3×C4C2×#6711yescharacteristic
220C2×C3×C4C2×#6711yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—21no—
223D12—41no—
224D12—41no—
225D12—41no—
226D12—41no—
227D12—81no—
228Dic6—21no—
229Dic6—21no—
230C3×C8—32no—
231C3×C8—61no—
232C3×C8—121no—
233C3⋊C8—21no—
234C3⋊C8—41no—
235C3⋊C8—121no—
236C3⋊C8—121no—
237C3⋊C8—121no—
238C3⋊C8—121no—
239C3⋊C8—241no—
240C3⋊C8—241no—
241C3⋊C8#21412yescomplemented
242C2×C32#46011yescharacteristic
243C2×C32#46811yescharacteristic
244C2×C32#47111yescharacteristic
245C3×D3—161no—
246C2×Q8—21no—
247C2×Q8—31no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251SD16—122no—
252SD16—122no—
253SD16—362no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—61no—
257C2×C8—181no—
258C2×C8—181no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C42—61no—
264C42C2×D3211yescharacteristic
265C22×C3—21no—
266C22×C3—241no—
267C22×C3—241no—
268C22×C3—241no—
269C22×C3#93811yescharacteristic
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—82no—
282C3×C4—81no—
283C3×C4—81no—
284C3×C4—121no—
285C3×C4—121no—
286C3×C4D3×SD1611yescharacteristic
287C3×C4#91411yescharacteristic
288C3×C4#91411yescharacteristic
289C3×C4#91511yescharacteristic
290C3×C4#93211yescharacteristic
291C3×C4C2×#21411yescharacteristic
292C3×C4#94211yescharacteristic
293C3×C4#94911yescharacteristic
294Dic3—41no—
295Dic3—41no—
296C32#168611yescharacteristic, complemented, Sylow
297C23—121no—
298Q8—41no—
299Q8—41no—
300Q8—61no—
301Q8—61no—
302D4—121no—
303D4—121no—
304D4—121no—
305D4—121no—
306D4—241no—
307C2×C4—41no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4#380511yescharacteristic
311C2×C4D3×#6711yescharacteristic
312C2×C4D3×#6711yescharacteristic
313C8—32no—
314C8—61no—
315C8—121no—
316C8—361no—
317C8—361no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—481no—
322C2×C3#504911yescharacteristic
323C2×C3#507511yescharacteristic
324C2×C3#516311yescharacteristic
325C2×C3#517411yescharacteristic
326C2×C3#517711yescharacteristic
327C2×C3#520311yescharacteristic
328D3—161no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22#6397011yescharacteristic
333C4—21no—
334C4—81no—
335C4—121no—
336C4—121no—
337C4D3×#21411yescharacteristic
338C4#6393411yescharacteristic
339C4#6393911yescharacteristic
340C4#6394611yescharacteristic
341C3—21no—
342C3#7112011yescharacteristic, complemented
343C3#7127311yescharacteristic, complemented
344C2—481no—
345C2#1059143811yescharacteristic
346C2#1059158711yescharacteristic
347C2#1059164211yescharacteristic
subgroupG/H
centerC22#63970
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. C3×#5094C2semidirect
4. #10590258C2semidirect
5. #10589413C2semidirect
6. #10589099C2semidirect
27. #5094D3semidirect
113. C3×C3⋊C8SD16semidirect
241. C3⋊C8#214semidirect
296. C32#1686semidirect
342. C3#71120semidirect
343. C3#71273semidirect

Maximal subgroups

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

subgroupindex
C3×#50942
C3×#50852
#105903642
#105902582
#105894132
#105890992
#105890902
#712733
#711203