Finite Group Databasebeta

← #11922956

Subgroups — #11922956

Order 1152.

696 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#64266C212yesmaximal, complemented
2C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#10593659C212yesmaximal, complemented
4C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
6#85407—33nomaximal
7#83579—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9#64266—24no—
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊#226C2212yes—
12C4⋊C4×C3⋊D3—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3⋊#859C2211yescharacteristic
15C2×#985—36no—
16C3×#550D311yescharacteristic, complemented
17C2×#950—32no—
18#5551—31no—
19C4⋊(#41⋊C3)D313yescomplemented
20#5425—36no—
21C4×GL(2,𝔽3)—33no—
22C4⋊#226—33no—
23#5239—32no—
24#5076—31no—
25#5072—31no—
26#5041—32no—
27#5037—32no—
28#5031—31no—
29C4×#215—31no—
30D4×C3⋊C8—31no—
31C2×C4×C3⋊D3—41no—
32C2×C4×C3⋊D3—42no—
33C3×#41⋊C3—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊GL(2,𝔽3)D412yescomplemented
36C3⋊#226—22no—
37C32×C4⋊C4—41no—
38C4⋊(C3⋊Dic3)—41no—
39#3750—42no—
40#3284—91noSylow
41C2×C3×#41C2×D312yes—
42C2×#235—32no—
43C2×#41⋊C3C2×D316yes—
44#985—612no—
45C2×GL(2,𝔽3)—33no—
46C2×#226—36no—
47C3×#127—31no—
48C3×#121—32no—
49C3×C4×Q8C2×D311yescharacteristic
50C3×C4×D4—31no—
51#950—64no—
52#950—64no—
53C2×#216—32no—
54C2×#215—31no—
55#938—61no—
56C2×#214—32no—
57C2×#214—32no—
58C2×#213—32no—
59#924—32no—
60C22×C3⋊C8—32no—
61D3×C4⋊C4—61no—
62D3×C4⋊C4—123no—
63#883—62no—
64C4×D12—31no—
65#869—31no—
66C4×SL(2,𝔽3)C2×D313yes—
67#859—33no—
68#835—31no—
69#832—32no—
70#830—32no—
71#810—32no—
72#809—31no—
73#808—31no—
74C4⋊(C3⋊C8)—31no—
75C4×C3⋊C8—31no—
76C22×C3⋊D3—41no—
77C2×C32×C4—41no—
78C2×C32×C4—42no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3—42no—
81C4×C3⋊D3—44no—
82C4×C3⋊D3—84no—
83C3×SL(2,𝔽3)C2×D411yescharacteristic
84C2×#135—92no—
85#550C3⋊D311yescharacteristic, complemented
86#547—91no—
87#492—91no—
88#465—92no—
89#462—91no—
90#459—91no—
91#452—92no—
92#449—92no—
93C4×SD16—91no—
94C8×D4—91no—
95C3×#41—22no—
96C3×#41—62no—
97C2×C3×Q8C22×D311yescharacteristic
98C2×C3×D4—32no—
99C2×C3×D4—31no—
100C2×C3×D4—61no—
101C22×C3×C4—32no—
102C2×#67—62no—
103#235—64no—
104#235—122no—
105C2×D12—31no—
106C2×C4×D3—61no—
107C2×C4×D3—62no—
108C2×C4×D3—123no—
109C2×C4×D3—126no—
110C2×Dic6—32no—
111#41⋊C3—26no—
112C2×SL(2,𝔽3)C22×D313yes—
113GL(2,𝔽3)—36no—
114#226—66no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—43no—
117C3×C4⋊C4S411yescharacteristic
118C3×C22⋊C4—32no—
119C3×C42—31no—
120#217—62no—
121#216—62no—
122#216—62no—
123#215—32no—
124#215—32no—
125#214—64no—
126#214—122no—
127#213—64no—
128(C2×D3)⋊C4—61no—
129C4⋊Dic3—31no—
130C4⋊Dic3—61no—
131C4⋊Dic3—61no—
132C4⋊Dic3—123no—
133Dic3⋊C4—62no—
134Dic3⋊C4—126no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—64no—
139C22×C32—41no—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C32×C4—42no—
144C32×C4—82no—
145C3⋊Dic3—42no—
146C3⋊Dic3—82no—
147C2×#41—92no—
148C2×#41C2×C3⋊D312yes—
149#135—184no—
150#135—184no—
151C2×Q16—92no—
152C2×SD16—91no—
153C2×SD16—92no—
154C2×SD16—92no—
155C2×D8—92no—
156C22×C8—92no—
157C4⋊Q8—91no—
158#127—31no—
159#123—182no—
160C22⋊Q8—92no—
161#121—32no—
162#121—181no—
163C4×Q8C2×C3⋊D311yescharacteristic
164C4×D4—31no—
165C4×D4—91no—
166C2×C4⋊C4—181no—
167#106—91no—
168C4⋊C8—91no—
169#103—92no—
170#103—91no—
171#102—91no—
172#102—92no—
173C22⋊C8—92no—
174C4×C8—91no—
175C23×C3—32no—
176C22×D3—61no—
177C22×D3—123no—
178C3×Q8—31no—
179C3×Q8D3×D411yescharacteristic
180C3×D4—62no—
181C3×D4—62no—
182C3×D4—62no—
183C3×D4—241no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—43no—
187C2×C3×C4—46no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4C2×S412yes—
191C2×C3×C4C2×S411yescharacteristic
192#67—124no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—123no—
196C2×Dic3—126no—
197D12—32no—
198D12—61no—
199C4×D3—124no—
200C4×D3—1212no—
201C4×D3—122no—
202C4×D3—2412no—
203Dic6—62no—
204Dic6—62no—
205SL(2,𝔽3)D3×D413yes—
206C3⋊C8—32no—
207C3⋊C8—62no—
208C3⋊C8—61no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C3⋊D3—44no—
213#41—22no—
214#41—62no—
215#41—184no—
216#41—362no—
217C2×Q8—92no—
218C2×Q8C22×C3⋊D311yescharacteristic
219C2×D4—32no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—91no—
223C2×D4—182no—
224C22×C4—32no—
225C22×C4—181no—
226C22×C4—182no—
227Q16—182no—
228Q16—182no—
229SD16—92no—
230SD16—92no—
231SD16—184no—
232SD16—362no—
233D8—184no—
234C2×C8—91no—
235C2×C8—91no—
236C2×C8—92no—
237C2×C8—184no—
238C4⋊C4—31no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C4⋊C4—181no—
242C4⋊C4—182no—
243C4⋊C4C3⋊S411yescharacteristic
244C22⋊C4—32no—
245C22⋊C4—181no—
246C22⋊C4—182no—
247C42—31no—
248C22×C3—43no—
249C22×C3—62no—
250C22×C3—62no—
251C22×C3—62no—
252C22×C3C22×S411yescharacteristic
253C2×D3—62no—
254C2×D3—126no—
255C2×D3—126no—
256C2×D3—126no—
257C2×D3—121no—
258C2×D3—121no—
259C3×C4—22no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—46no—
263C3×C4—61no—
264C3×C4—86no—
265C3×C4#98512yescomplemented
266Dic3—126no—
267Dic3—122no—
268Dic3—121no—
269Dic3—246no—
270C32—41noSylow
271C23—32no—
272C23—181no—
273Q8—31no—
274Q8—182no—
275Q8—182no—
276Q8D4×C3⋊D311yescharacteristic
277D4—62no—
278D4—62no—
279D4—62no—
280D4—92no—
281D4—181no—
282D4—241no—
283D4—364no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—364no—
291C2×C4—362no—
292C2×C4C2×C3⋊S412yes—
293C2×C4C2×C3⋊S411yescharacteristic
294C8—92no—
295C8—182no—
296C8—181no—
297C2×C3—43no—
298C2×C3—43no—
299C2×C3—43no—
300C2×C3—122no—
301C2×C3D4×S411yescharacteristic
302C2×C3C2×#98511yescharacteristic
303C2×C3#592411yescharacteristic
304D3—1212no—
305D3—122no—
306C22—62no—
307C22—62no—
308C22—62no—
309C22—182no—
310C22—361no—
311C22—361no—
312C22C22×C3⋊S411yescharacteristic
313C4—22no—
314C4—31no—
315C4—31no—
316C4—61no—
317C4—362no—
318C4—361no—
319C4#6426612yescomplemented
320C3—43no—
321C3#8540711yescharacteristic, complemented
322C2—122no—
323C2—362no—
324C2D4×C3⋊S411yescharacteristic
325C2C2×#6426611yescharacteristic
326C2#1059658711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#550D3
socleC22×C3C22×S4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64266C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. #10593659C2semidirect
4. C4×C3⋊GL(2,𝔽3)C2semidirect
5. C4⋊(C3⋊#226)C2semidirect
16. C3×#550D3semidirect
19. C4⋊(#41⋊C3)D3semidirect
35. C3⋊GL(2,𝔽3)D4semidirect
85. #550C3⋊D3semidirect
265. C3×C4#985semidirect
319. C4#64266semidirect
321. C3#85407semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
C3×C4⋊(#41⋊C3)2
#105936592
C4×C3⋊GL(2,𝔽3)2
C4⋊(C3⋊#226)2
#854073
#835793
C4⋊C4×C3⋊D34