Finite Group Databasebeta

← #11923088

Subgroups — #11923088

Order 1152.

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

normal subgroups only

This group has 864 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
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#10593661C212yesmaximal, complemented
4#10593655C212yesmaximal, complemented
5#10593654C211yesmaximal, characteristic, complemented
6#85537—33nomaximal
7#84113—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic, complemented
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C3⋊GL(2,𝔽3)—22no—
13C2×C3⋊GL(2,𝔽3)C2212yes—
14#64146—41nomaximal
15C3⋊#859C2212yescomplemented
16C2×C3×2−1+4D311yescharacteristic, complemented
17C2×2−1+4⋊C3D313yescomplemented
18C22×GL(2,𝔽3)—33no—
19#5830—31no—
20C2×#949—31no—
21C22×#215—31no—
22#5427—36no—
23#5421—36no—
24#5420—33no—
25#5238—32no—
26#5238—32no—
27#5234—31no—
28#5233—31no—
29#5227—32no—
30#5227—32no—
31#5225—31no—
32C23×C3⋊D3—41no—
33C2×C32×D4—41no—
34C2×#638—42no—
35C3×#41⋊C3—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)D412yescomplemented
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3⋊GL(2,𝔽3)—22no—
40#3757—41no—
41#3752—42no—
42#3010—91noSylow
43C3×2−1+4—22no—
44C3×2−1+4—62no—
45C2×C3×#41—31no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41C2×D311yescharacteristic, complemented
49C22×C3×Q8—31no—
50C22×C3×Q8C2×D311yescharacteristic
51C22×C3×Q8C2×D311yescharacteristic, complemented
52C22×D12—31no—
532−1+4⋊C3—26no—
54C2×#41⋊C3C2×D313yescomplemented
55C22×SL(2,𝔽3)C2×D313yes—
56C22×SL(2,𝔽3)C2×D313yescomplemented
57C2×GL(2,𝔽3)—36no—
58C2×GL(2,𝔽3)—66no—
59#949—64no—
60#947—61no—
61C2×#215—32no—
62C2×#215—32no—
63C2×#215—32no—
64C2×#215—62no—
65C2×#215—62no—
66#940—61no—
67#937—61no—
68#937—61no—
69#937—123no—
70C2×#213—32no—
71#930—32no—
72#930—32no—
73#926—31no—
74C2×#208—31no—
75C22×C3⋊C8—31no—
76#859—36no—
77#835—32no—
78#835—32no—
79#835—32no—
80#832—32no—
81#830—32no—
82#830—32no—
83C23×C32—41no—
84C23×C32—41no—
85C22×C3⋊D3—42no—
86C22×C3⋊D3—82no—
87C22×C3⋊D3—84no—
88C32×D4—82no—
89C2×C32×C4—41no—
90#638—84no—
91C2×C3⋊Dic3—42no—
92C3×SL(2,𝔽3)C2×D411yescharacteristic
93C2×2−1+4C3⋊D311yescharacteristic, complemented
94C2×AGL(1,ℤ/8)—91no—
95C22×SD16—91no—
96#535—91no—
97#449—92no—
98#449—92no—
99#448—92no—
100#448—92no—
101#419—91no—
102#416—91no—
103#408—91no—
104C23×D3—61no—
105C23×D3—123no—
106C3×#41—21no—
107C3×#41—34no—
108C3×#41—62no—
109C3×#41—62no—
110C3×#41—62no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—61no—
114C2×C3×Q8—21no—
115C2×C3×Q8—31no—
116C2×C3×Q8—31no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8—62no—
120C2×C3×Q8—62no—
121C2×C3×Q8—61no—
122C2×C3×Q8#6712yescomplemented
123C2×C3×Q8C22×D311yescharacteristic
124C2×C3×D4—31no—
125C2×C3×D4—31no—
126C2×C3×D4—31no—
127C2×C3×D4—43no—
128C2×C3×D4S411yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C2×#67—62no—
134C2×#67—62no—
135C2×#67—126no—
136C2×D12—32no—
137C2×D12—62no—
138C2×D12—62no—
139C2×D12—61no—
140#41⋊C3—23no—
141C2×SL(2,𝔽3)—23no—
142C2×SL(2,𝔽3)#6716yescomplemented
143C2×SL(2,𝔽3)C22×D313yes—
144GL(2,𝔽3)—66no—
145#217—61no—
146#217—123no—
147#215—64no—
148#215—62no—
149#215—64no—
150#215—62no—
151#213—64no—
152(C2×D3)⋊C4—62no—
153(C2×D3)⋊C4—62no—
154(C2×D3)⋊C4—126no—
155C4⋊Dic3—32no—
156C4×Dic3—61no—
157#208—62no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—62no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—82no—
164C22×C32—82no—
165C22×C32—81no—
166C2×C3⋊D3—44no—
167C2×C3⋊D3—82no—
168C2×C3⋊D3—82no—
169C2×C3⋊D3—88no—
170C32×C4—81no—
171C3⋊Dic3—82no—
1722−1+4—22no—
1732−1+4—62no—
174C2×#41—31no—
175C2×#41—31no—
176C2×#41—31no—
177C2×#41C2×C3⋊D311yescharacteristic, complemented
178C22×Q8—31no—
179C22×Q8C2×C3⋊D311yescharacteristic
180C22×Q8C2×C3⋊D311yescharacteristic, complemented
181C22×D4—91no—
182AGL(1,ℤ/8)—184no—
183C2×SD16—92no—
184C2×SD16—92no—
185C2×SD16—92no—
186C2×SD16—182no—
187C2×SD16—182no—
188C2×D8—92no—
189C2×M16—91no—
190C22×C8—91no—
191#127—181no—
192#124—181no—
193#121—92no—
194#121—92no—
195#120—181no—
196#120—181no—
197#117—91no—
198#103—92no—
199#103—92no—
200#103—92no—
201#102—92no—
202C22⋊C8—92no—
203C22⋊C8—92no—
204C23×C3—31no—
205C23×C3—43no—
206C23×C3—43no—
207C23×C3C2×S411yescharacteristic
208C23×C3C2×S411yescharacteristic
209C22×D3—62no—
210C22×D3—126no—
211C22×D3—122no—
212C22×D3—246no—
213C22×D3—242no—
214C22×D3—2412no—
215C3×Q8—32no—
216C3×Q8—31no—
217C3×Q8—32no—
218C3×Q8—62no—
219C3×Q8—61no—
220C3×Q8—61no—
221C3×Q8C2×#6711yescharacteristic
222C3×D4—22no—
223C3×D4—32no—
224C3×D4—62no—
225C3×D4—61no—
226C3×D4—62no—
227C3×D4—86no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—43no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—64no—
239C2×C3×C4C2×S411yescharacteristic
240#67—242no—
241#67—242no—
242#67—2412no—
243C2×Dic3—62no—
244C2×Dic3—126no—
245D12—62no—
246D12—62no—
247D12—122no—
248SL(2,𝔽3)C2×#6713yes—
249C3⋊C8—62no—
250C3⋊C8—62no—
251C2×C32—42no—
252C2×C32—41no—
253C2×C32—82no—
254C2×C32—161no—
255C3⋊D3—84no—
256C24—181no—
257#41—21no—
258#41—34no—
259#41—62no—
260#41—62no—
261#41—62no—
262#41—62no—
263#41—62no—
264#41—61no—
265C2×Q8—21no—
266C2×Q8—31no—
267C2×Q8—31no—
268C2×Q8—31no—
269C2×Q8—32no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8—61no—
273C2×Q8#63812yescomplemented
274C2×Q8C22×C3⋊D311yescharacteristic
275C2×D4—31no—
276C2×D4—31no—
277C2×D4—31no—
278C2×D4—92no—
279C2×D4—182no—
280C2×D4—182no—
281C2×D4—181no—
282C2×D4—182no—
283C2×D4—182no—
284C2×D4C3⋊S411yescharacteristic
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—61no—
289SD16—184no—
290SD16—182no—
291SD16—184no—
292SD16—182no—
293D8—184no—
294M16—182no—
295C2×C8—92no—
296C2×C8—92no—
297C2×C8—182no—
298C4⋊C4—92no—
299C22⋊C4—182no—
300C22⋊C4—181no—
301C22⋊C4—182no—
302C42—181no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—46no—
307C22×C3—43no—
308C22×C3—62no—
309C22×C3—61no—
310C22×C3—86no—
311C22×C3—86no—
312C22×C3—83no—
313C22×C3GL(2,ℤ/4)12yes—
314C22×C3C22×S411yescharacteristic
315C2×D3—1212no—
316C2×D3—122no—
317C2×D3—122no—
318C2×D3—122no—
319C2×D3—246no—
320C2×D3—246no—
321C2×D3—244no—
322C2×D3—2424no—
323C3×C4—21no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—32no—
327C3×C4—61no—
328C3×C4—83no—
329Dic3—122no—
330Dic3—246no—
331C32—41noSylow
332C23—31no—
333C23—182no—
334C23—362no—
335C23—722no—
336C23C2×C3⋊S411yescharacteristic
337C23C2×C3⋊S411yescharacteristic
338Q8—32no—
339Q8—31no—
340Q8—32no—
341Q8—62no—
342Q8—61no—
343Q8—61no—
344Q8C2×#63811yescharacteristic
345D4—22no—
346D4—32no—
347D4—62no—
348D4—61no—
349D4—62no—
350D4—182no—
351D4—182no—
352D4—362no—
353D4—722no—
354D4—722no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—64no—
365C2×C4—182no—
366C2×C4C2×C3⋊S411yescharacteristic
367C8—182no—
368C8—182no—
369C2×C3—22no—
370C2×C3—41no—
371C2×C3—46no—
372C2×C3—43no—
373C2×C3—86no—
374C2×C3—121no—
375C2×C3—163no—
376C2×C3#592612yes—
377C2×C3C2×GL(2,ℤ/4)11yescharacteristic
378D3—2412no—
379D3—242no—
380C22—22no—
381C22—22no—
382C22—21no—
383C22—62no—
384C22—61no—
385C22—362no—
386C22—362no—
387C22—362no—
388C22—724no—
389C22#6426912yes—
390C22C22×C3⋊S411yescharacteristic
391C4—21no—
392C4—31no—
393C4—31no—
394C4—32no—
395C4—61no—
396C4—362no—
397C3—43no—
398C3#8553711yescharacteristic, complemented
399C2—22no—
400C2—41no—
401C2—121no—
402C2—722no—
403C2#1059658912yes—
404C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C22×C3⋊GL(2,𝔽3)C2semidirect
3. #10593661C2semidirect
4. #10593655C2semidirect
5. #10593654C2semidirect
9. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
15. C3⋊#859C22semidirect
16. C2×C3×2−1+4D3semidirect
17. C2×2−1+4⋊C3D3semidirect
37. C2×C3×SL(2,𝔽3)D4semidirect
48. C2×C3×#41C2×D3semidirect
51. C22×C3×Q8C2×D3semidirect
54. C2×#41⋊C3C2×D3semidirect
56. C22×SL(2,𝔽3)C2×D3semidirect
93. C2×2−1+4C3⋊D3semidirect
122. C2×C3×Q8#67semidirect
142. C2×SL(2,𝔽3)#67semidirect
177. C2×#41C2×C3⋊D3semidirect
180. C22×Q8C2×C3⋊D3semidirect
273. C2×Q8#638semidirect
398. C3#85537semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3⋊GL(2,𝔽3)2
#105936612
#105936552
#105936542
#855373
#841133
#641464