Finite Group Databasebeta

← #11922915

Subgroups — #11922915

Order 1152.

660 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 89 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×#64266C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593628C212yesmaximal, complemented
5C4×C3⋊#226C212yesmaximal, complemented
6#85365—33nomaximal
7#82557—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64266C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊#226C2211yescharacteristic
12#64101—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yes—
14C3⋊#859C2212yes—
15C2×#985—33no—
16C3×#520D311yescharacteristic, complemented
17C2×#950—31no—
18#5523—31no—
19#520⋊C3D313yescomplemented
20#5426—33no—
21#5394—36no—
22C4×#226—36no—
23#5232—31no—
24C4×#216—32no—
25#5026—32no—
26#5018—32no—
27C4×#213—32no—
28#4997—31no—
29#4966—31no—
30C2×#807—31no—
31C2×C4×C3⋊D3—41no—
32C3×#41⋊C3C2×C412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊GL(2,𝔽3)C2×C412yescomplemented
35C3⋊#226C2×C412yescomplemented
36C32×C4⋊C4—41no—
37#3752—42no—
38C4⋊(C3⋊Dic3)—41no—
39C4×C3⋊Dic3—42no—
40#2951—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C2×#235—31no—
43C2×#41⋊C3C2×D313yes—
44#985—312no—
45C2×GL(2,𝔽3)—33no—
46C2×#226—33no—
47C3×C4×Q8C2×D312yes—
48C3×C4×D4—32no—
49C3×#117—31no—
50C2×C3×C4⋊C4—31no—
51#950—34no—
52#950—34no—
53C2×#216—31no—
54C2×#215—31no—
55C2×#214—31no—
56C2×#213—31no—
57C2×C4⋊Dic3—31no—
58C2×#208—31no—
59C22×C3⋊C8—31no—
60#892—61no—
61#892—123no—
62D3×C4⋊C4—61no—
63#67⋊C4—62no—
64C4×D12—32no—
65C4×Dic6—32no—
66C4×SL(2,𝔽3)C2×D316yes—
67#859—36no—
68#835—32no—
69#832—32no—
70#810—32no—
71#809—32no—
72#807—32no—
73#807—32no—
74#803—32no—
75C4×C3⋊C8—32no—
76C22×C3⋊D3—41no—
77C2×C32×C4—41no—
78C2×C32×C4—42no—
79C2×C3⋊Dic3—41no—
80C2×C3⋊Dic3—42no—
81C4×C3⋊D3—44no—
82C3×SL(2,𝔽3)C22×C411yescharacteristic
83C2×#135—91no—
84#520—91no—
85#520C3⋊D311yescharacteristic, complemented
86#440—92no—
87#440—92no—
88C4×Q16—92no—
89C4×D8—92no—
90C2×#107—91no—
91#417—91no—
92#417—91no—
93#405—91no—
94C3×#41—32no—
95C3×#41C4×D312yes—
96C2×C3×Q8C22×D311yescharacteristic
97C2×C3×D4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—61no—
100C2×#67—61no—
101C22×Dic3—61no—
102#235—34no—
103#235—62no—
104C2×D12—31no—
105C2×C4×D3—61no—
106C2×C4×D3—62no—
107C2×C4×D3—123no—
108C2×Dic6—31no—
109#41⋊C3C4×D316yescomplemented
110C2×SL(2,𝔽3)C22×D313yes—
111GL(2,𝔽3)—36no—
112#226—36no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—43no—
116C3×C4⋊C4S411yescharacteristic
117C3×C22⋊C4—32no—
118C3×C42—32no—
119#216—32no—
120#216—32no—
121#215—32no—
122#215—32no—
123#214—34no—
124#213—34no—
125(C2×D3)⋊C4—62no—
126(C2×D3)⋊C4—126no—
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3—123no—
130Dic3⋊C4—62no—
131C4×Dic3—62no—
132C4×Dic3—126no—
133#208—62no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—34no—
137C2×C3⋊C8—32no—
138C22×C32—41no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—81no—
141C2×C3⋊D3—81no—
142C32×C4—42no—
143C32×C4—82no—
144C3⋊Dic3—42no—
145C3⋊Dic3—82no—
146C2×#41—91no—
147C2×#41C2×C3⋊D311yescharacteristic
148#135—94no—
149#135—94no—
150C2×Q16—91no—
151C2×SD16—91no—
152C2×SD16—91no—
153C2×D8—91no—
154C2×M16—91no—
155C22×C8—91no—
156C4×Q8—92no—
157C4×Q8C2×C3⋊D312yes—
158C4×D4—32no—
159C4×D4—92no—
160C4×D4—182no—
161#117—31no—
162#117—181no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—91no—
165C2×C4⋊C4—181no—
166#107—92no—
167#107—92no—
168#103—92no—
169#103—92no—
170#102—92no—
171#102—92no—
172#97—92no—
173C4×C8—92no—
174C23×C3—31no—
175C22×D3—61no—
176C22×D3—123no—
177C3×Q8—31no—
178C3×Q8C2×C4×D311yescharacteristic
179C3×D4—32no—
180C3×D4—61no—
181C2×C3×C4—31no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—43no—
186C2×C3×C4—46no—
187C2×C3×C4—241no—
188C2×C3×C4C2×S412yes—
189C2×C3×C4C2×S411yescharacteristic
190#67—64no—
191C2×Dic3—61no—
192C2×Dic3—62no—
193C2×Dic3—123no—
194C2×Dic3—126no—
195C2×Dic3—241no—
196D12—32no—
197D12—61no—
198C4×D3—64no—
199C4×D3—1212no—
200C4×D3—242no—
201Dic6—32no—
202Dic6—61no—
203SL(2,𝔽3)C2×C4×D313yes—
204C3⋊C8—32no—
205C3⋊C8—32no—
206C3⋊C8—62no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C3⋊D3—82no—
211#41—32no—
212#41—94no—
213#41—182no—
214#41C4×C3⋊D312yes—
215C2×Q8—91no—
216C2×Q8C22×C3⋊D311yescharacteristic
217C2×D4—31no—
218C2×D4—91no—
219C2×D4—181no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—182no—
223C22×C4—181no—
224C22×C4—181no—
225Q16—92no—
226Q16—92no—
227SD16—94no—
228SD16—92no—
229SD16—92no—
230D8—94no—
231M16—182no—
232C2×C8—91no—
233C2×C8—91no—
234C2×C8—94no—
235C2×C8—92no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—92no—
239C4⋊C4—182no—
240C4⋊C4—181no—
241C4⋊C4C3⋊S411yescharacteristic
242C22⋊C4—32no—
243C22⋊C4—182no—
244C42—32no—
245C42—182no—
246C22×C3—32no—
247C22×C3—43no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3C22×S411yescharacteristic
251C2×D3—62no—
252C2×D3—126no—
253C2×D3—121no—
254C2×D3—121no—
255C2×D3—243no—
256C2×D3—243no—
257C3×C4—22no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—46no—
261C3×C4—62no—
262C3×C4—86no—
263C3×C4C4×S412yes—
264Dic3—62no—
265Dic3—126no—
266Dic3—122no—
267Dic3—246no—
268C32—41noSylow
269C23—31no—
270C23—181no—
271Q8—31no—
272Q8—92no—
273Q8—181no—
274Q8C2×C4×C3⋊D311yescharacteristic
275D4—32no—
276D4—61no—
277D4—92no—
278D4—181no—
279D4—184no—
280C2×C4—31no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—181no—
285C2×C4—184no—
286C2×C4—182no—
287C2×C4—241no—
288C2×C4—722no—
289C2×C4—721no—
290C2×C4C2×C3⋊S412yes—
291C2×C4C2×C3⋊S411yescharacteristic
292C8—92no—
293C8—92no—
294C8—182no—
295C2×C3—43no—
296C2×C3—43no—
297C2×C3—43no—
298C2×C3—62no—
299C2×C3C2×C4×S411yescharacteristic
300C2×C3#592511yescharacteristic
301C2×C3#592711yescharacteristic
302D3—122no—
303D3—246no—
304C22—32no—
305C22—61no—
306C22—61no—
307C22—182no—
308C22—361no—
309C22—361no—
310C22C22×C3⋊S411yescharacteristic
311C4—22no—
312C4—31no—
313C4—31no—
314C4—62no—
315C4—182no—
316C4—362no—
317C4C4×C3⋊S412yes—
318C3—43no—
319C3#8536511yescharacteristic, complemented
320C2—62no—
321C2—362no—
322C2C2×C4×C3⋊S411yescharacteristic
323C2#1059658811yescharacteristic
324C2#1059659011yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC3×#520D3
socleC22×C3C22×S4
prime radicalC2×#64266C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC4⋊C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. #10593660C2semidirect
4. #10593628C2semidirect
5. C4×C3⋊#226C2semidirect
9. #64266C4semidirect
16. C3×#520D3semidirect
19. #520⋊C3D3semidirect
32. C3×#41⋊C3C2×C4semidirect
34. C3⋊GL(2,𝔽3)C2×C4semidirect
35. C3⋊#226C2×C4semidirect
85. #520C3⋊D3semidirect
109. #41⋊C3C4×D3semidirect
319. C3#85365semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
C3×#520⋊C32
#105936602
#105936282
C4×C3⋊#2262
#853653
#825573
#641014