Finite Group Databasebeta

← #11923071

Subgroups — #11923071

Order 1152.

754 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#64264C211yesmaximal, characteristic, complemented
3#10593661C212yesmaximal, complemented
4#10593659C212yesmaximal, complemented
5#10593654C211yesmaximal, characteristic, complemented
6#85522—33nomaximal
7#84081—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2212yes—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#64264—22no—
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊#226C2211yescharacteristic, complemented
14#64138—41nomaximal
15C3⋊#859C2212yescomplemented
16C22×C3×#41D311yescharacteristic, complemented
17C22×#41⋊C3D313yescomplemented
18C2×#983—33no—
19C2×#942—31no—
20C2×#932—31no—
21#5791—31no—
22#5427—36no—
23#5425—36no—
24#5420—33no—
25#5239—32no—
26#5239—32no—
27#5238—32no—
28#5238—32no—
29#5225—31no—
30#5215—31no—
31#5207—31no—
32C22×C32×C4—41no—
33C2×#638—41no—
34C3×#41⋊C3—22no—
35C2×C3×SL(2,𝔽3)D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C3⋊GL(2,𝔽3)—21no—
38C3⋊#226—21no—
39#3757—41no—
40#3752—42no—
41#3750—42no—
42#3005—91noSylow
43C2×C3×#41—22no—
44C2×C3×#41—32no—
45C2×C3×#41—62no—
46C2×C3×#41C2×D312yes—
47C22×C3×Q8C2×D311yescharacteristic
48C22×C3×D4—31no—
49C23×C3×C4—31no—
50C2×#235—31no—
51C2×#41⋊C3—26no—
52C2×#41⋊C3C2×D316yes—
53C22×SL(2,𝔽3)C2×D313yes—
54#983—66no—
55C2×GL(2,𝔽3)—33no—
56C2×#226—33no—
57#953—61no—
58C2×#216—31no—
59C2×#216—31no—
60#942—62no—
61#942—62no—
62C2×#215—31no—
63C2×#215—31no—
64C2×#214—32no—
65#932—64no—
66C2×#213—32no—
67#930—32no—
68#929—61no—
69#929—123no—
70C4×#67—62no—
71#926—31no—
72#924—32no—
73C2×#208—31no—
74C2×#208—31no—
75#859—36no—
76#835—32no—
77#835—32no—
78#832—32no—
79#832—32no—
80#830—32no—
81#830—32no—
82C23×C32—41no—
83C22×C3⋊D3—41no—
84C2×C32×C4—42no—
85C2×C32×C4—82no—
86#638—82no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—42no—
89C3×SL(2,𝔽3)C2×D411yescharacteristic
90C22×#41C3⋊D311yescharacteristic, complemented
91C2×#137—91no—
92C2×AGL(1,ℤ/8)—91no—
93#525—91no—
94#452—92no—
95#452—92no—
96#449—92no—
97#449—92no—
98#419—91no—
99#418—91no—
100#407—91no—
101C24×C3—31no—
102C3×#41—22no—
103C3×#41—64no—
104C3×#41—62no—
105C3×#41—64no—
106C2×C3×Q8—31no—
107C2×C3×Q8—32no—
108C2×C3×Q8#6712yes—
109C2×C3×Q8C22×D311yescharacteristic
110C2×C3×D4—32no—
111C2×C3×D4—34no—
112C2×C3×D4—61no—
113C2×C3×D4—62no—
114C22×C3×C4—31no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—43no—
118C22×C3×C4—62no—
119C22×C3×C4—62no—
120C22×C3×C4S411yescharacteristic
121C2×#67—62no—
122C2×#67—61no—
123C2×#67—123no—
124#235—62no—
125#235—62no—
126C2×D12—31no—
127C2×C4×D3—61no—
128C2×Dic6—31no—
129#41⋊C3—26no—
130C2×SL(2,𝔽3)#6716yescomplemented
131C2×SL(2,𝔽3)C22×D313yes—
132GL(2,𝔽3)—63no—
133#226—63no—
134#217—62no—
135#217—61no—
136#217—123no—
137#216—62no—
138#216—61no—
139#216—61no—
140#215—62no—
141#215—61no—
142#215—61no—
143#214—64no—
144#213—64no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—126no—
147C4⋊Dic3—32no—
148Dic3⋊C4—62no—
149Dic3⋊C4—126no—
150C4×Dic3—61no—
151#208—62no—
152#208—62no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—32no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—82no—
159C2×C3⋊D3—81no—
160C2×C3⋊D3—82no—
161C32×C4—82no—
162C3⋊Dic3—82no—
163C3⋊Dic3—81no—
164C2×#41—22no—
165C2×#41—32no—
166C2×#41—62no—
167C2×#41—91no—
168C2×#41C2×C3⋊D312yes—
169C22×Q8C2×C3⋊D311yescharacteristic
170C22×D4—31no—
171C23×C4—31no—
172#137—182no—
173#137—182no—
174AGL(1,ℤ/8)—184no—
175C2×Q16—91no—
176C2×Q16—91no—
177C2×SD16—92no—
178C2×SD16—91no—
179C2×SD16—91no—
180C2×D8—92no—
181C2×M16—91no—
182C2×M16—91no—
183#123—181no—
184C22⋊Q8—92no—
185#121—92no—
186#120—181no—
187C4×D4—182no—
188#117—91no—
189#103—92no—
190#103—92no—
191#102—92no—
192#102—92no—
193C22⋊C8—92no—
194C22⋊C8—92no—
195C23×C3—32no—
196C23×C3—43no—
197C23×C3—62no—
198C23×C3—122no—
199C23×C3C2×S411yescharacteristic
200C22×D3—61no—
201C22×D3—123no—
202C3×Q8—32no—
203C3×Q8—31no—
204C3×Q8—61no—
205C3×Q8C2×#6711yescharacteristic
206C3×D4—34no—
207C3×D4—62no—
208C3×D4—64no—
209C2×C3×C4—22no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—46no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—64no—
218C2×C3×C4—86no—
219C2×C3×C4C2×S412yes—
220#67—124no—
221#67—122no—
222#67—246no—
223C2×Dic3—62no—
224C2×Dic3—61no—
225C2×Dic3—126no—
226C2×Dic3—123no—
227D12—61no—
228D12—61no—
229C4×D3—122no—
230Dic6—61no—
231Dic6—61no—
232SL(2,𝔽3)C2×#6713yes—
233C3⋊C8—62no—
234C3⋊C8—61no—
235C3⋊C8—61no—
236C2×C32—42no—
237C2×C32—41no—
238C2×C32—82no—
239C3⋊D3—161no—
240C24—31no—
241#41—22no—
242#41—64no—
243#41—62no—
244#41—64no—
245#41—182no—
246#41—182no—
247C2×Q8—31no—
248C2×Q8—32no—
249C2×Q8—91no—
250C2×Q8#63812yes—
251C2×Q8C22×C3⋊D311yescharacteristic
252C2×D4—32no—
253C2×D4—34no—
254C2×D4—61no—
255C2×D4—62no—
256C2×D4—91no—
257C2×D4—182no—
258C2×D4—181no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—181no—
265C22×C4C3⋊S411yescharacteristic
266Q16—182no—
267Q16—181no—
268Q16—181no—
269SD16—184no—
270SD16—182no—
271SD16—181no—
272SD16—181no—
273D8—184no—
274M16—182no—
275M16—182no—
276C2×C8—91no—
277C2×C8—91no—
278C2×C8—92no—
279C4⋊C4—92no—
280C4⋊C4—182no—
281C22⋊C4—182no—
282C22⋊C4—181no—
283C22⋊C4—182no—
284C42—181no—
285C22×C3—22no—
286C22×C3—46no—
287C22×C3—43no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—86no—
292C22×C3—124no—
293C22×C3GL(2,ℤ/4)12yes—
294C22×C3C22×S411yescharacteristic
295C2×D3—122no—
296C2×D3—121no—
297C2×D3—243no—
298C2×D3—246no—
299C3×C4—22no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—32no—
303C3×C4—86no—
304Dic3—122no—
305Dic3—121no—
306Dic3—243no—
307Dic3—246no—
308C32—41noSylow
309C23—32no—
310C23—62no—
311C23—122no—
312C23—181no—
313C23C2×C3⋊S411yescharacteristic
314Q8—32no—
315Q8—31no—
316Q8—61no—
317Q8—181no—
318Q8—181no—
319Q8C2×#63811yescharacteristic
320D4—34no—
321D4—62no—
322D4—64no—
323D4—181no—
324D4—181no—
325D4—364no—
326D4—362no—
327C2×C4—22no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—64no—
335C2×C4—182no—
336C2×C4—181no—
337C2×C4—362no—
338C2×C4C2×C3⋊S412yes—
339C8—182no—
340C8—181no—
341C8—181no—
342C2×C3—22no—
343C2×C3—46no—
344C2×C3—43no—
345C2×C3—86no—
346C2×C3—122no—
347C2×C3#592312yes—
348C2×C3C2×GL(2,ℤ/4)11yescharacteristic
349D3—241no—
350D3—483no—
351C22—22no—
352C22—62no—
353C22—62no—
354C22—62no—
355C22—124no—
356C22—362no—
357C22—361no—
358C22#6426912yes—
359C22C22×C3⋊S411yescharacteristic
360C4—22no—
361C4—31no—
362C4—31no—
363C4—32no—
364C4—362no—
365C4—361no—
366C3—43no—
367C3#8552211yescharacteristic, complemented
368C2—22no—
369C2—122no—
370C2—721no—
371C2#1059658612yes—
372C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×#41D3
socleC22×C3C22×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C2×#64264C2semidirect
3. #10593661C2semidirect
4. #10593659C2semidirect
5. #10593654C2semidirect
12. C2×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3⋊#226C22semidirect
15. C3⋊#859C22semidirect
16. C22×C3×#41D3semidirect
17. C22×#41⋊C3D3semidirect
35. C2×C3×SL(2,𝔽3)D4semidirect
90. C22×#41C3⋊D3semidirect
130. C2×SL(2,𝔽3)#67semidirect
367. C3#85522semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×#642642
#105936612
#105936592
#105936542
#855223
#840813
#641384