Finite Group Databasebeta

← #11808242

Subgroups — #11808242

Order 1152.

2333 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 2333 conjugacy classes of subgroups — too many to draw. Only its 232 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×#3832C211yesmaximal, characteristic, complemented
2C2×#63881C213yesmaximal, complemented
3C3×#5231C211yesmaximal, characteristic, complemented
4C3×#5230C211yesmaximal, characteristic, complemented
5#10591395C213yesmaximal, complemented
6#10591321C213yesmaximal, complemented
7#10591260C213yesmaximal, complemented
8#77169—31nomaximal
9#76777—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3832—212no—
12C22×C4⋊(C3⋊D3)C2213yes—
13C22×C4×C3⋊D3C2213yes—
14C22×#626C2213yes—
15C22×D3×Dic3C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4C2213yes—
17C2×C3×Dic3⋊C4C2213yes—
18C2×#3722C2216yes—
19C2×#3722C2213yes—
20C2×Dic32C2213yes—
21#63881—212no—
22C3×C3⋊#95C2213yes—
23C3×C3⋊#95C2213yes—
24C22×#239—31no—
25C22×#239—31no—
26C22×#239—61no—
27C2×#893—33no—
28C2×#67⋊C4—33no—
29C3×#395—31no—
30C3×#393—31no—
31#5231D311yescharacteristic, complemented
32#5230D311yescharacteristic, complemented
33#4991—33no—
34#4960—33no—
35#4672—33no—
36#4671—33no—
37#4666—33no—
38#4666—33no—
39C23×C3⋊D3C2313yes—
40C23×C3×D3C2311yescharacteristic
41C2×C32×Q8—26no—
42C22×C32×C4C2313yes—
43C22×C3⋊Dic3C2311yescharacteristic
44#3832—232no—
45C2×C4⋊(C3⋊D3)—212no—
46C2×C4⋊(C3⋊D3)—26no—
47C2×C4×C3⋊D3—212no—
48C2×C4×C3⋊D3D4112yescomplemented
49C22×C3×Dic3C2311yescharacteristic
50C22×C3×Dic3C2313yes—
51C22×C3×Dic3C2313yes—
52C2×#626—26no—
53C2×#626—26no—
54C2×#626—26no—
55C2×D3×Dic3—26no—
56C3×(C2×D3)⋊C4—26no—
57C3×Dic3⋊C4—26no—
58#3722—26no—
59#3722—212no—
60Dic32—26no—
61#2388—91noSylow
62C22×C3×Q8—21no—
63C22×C3×Q8C2×D311yescharacteristic
64C22×C3×Q8C2×D311yescharacteristic
65C22×#67—33no—
66C23×Dic3—31no—
67C2×#239—612no—
68C2×#239—612no—
69C2×#239—624no—
70C22×D12—33no—
71C22×D12—33no—
72C22×D12—33no—
73C22×D12—63no—
74C22×C4×D3—33no—
75C22×C4×D3—33no—
76C22×C4×D3—31no—
77C22×C4×D3—63no—
78C2×C3×C4⋊C4—33no—
79C2×C3×C22⋊C4—33no—
80C2×(C2×D3)⋊C4—36no—
81C2×(C2×D3)⋊C4—36no—
82C2×(C2×D3)⋊C4—33no—
83C2×(C2×D3)⋊C4—33no—
84C2×(C2×D3)⋊C4C2×D313yes—
85C2×Dic3⋊C4C2×D313yes—
86C2×C4×Dic3—33no—
87C2×C4×Dic3—33no—
88#893—612no—
89#67⋊C4—612no—
90C3×#95—33no—
91C3×#95—33no—
92C3⋊#95C2×D313yes—
93C3⋊#95C2×D313yes—
94C23×C32C2411yescharacteristic
95C22×C3⋊D3—26no—
96C22×C3⋊D3—26no—
97C22×C3⋊D3—23no—
98C22×C3⋊D3—23no—
99C22×C3⋊D3C2×D416yes—
100C22×C3×D3—23no—
101C22×C3×D3—23no—
102C22×C3×D3#4112yescomplemented
103C32×Q8—28no—
104C2×C32×C4—26no—
105C2×C32×C4C2×D416yes—
106C2×C3⋊Dic3C2×D416yes—
107C4⋊(C3⋊D3)—224no—
108C4×C3⋊D3—224no—
109C2×C3×Dic3—23no—
110C2×C3×Dic3—23no—
111C2×C3×Dic3—23no—
112C2×C3×Dic3—23no—
113C2×C3×Dic3—26no—
114C2×C3×Dic3#4116yescomplemented
115#626—224no—
116D3×Dic3—44no—
117C22×#41—91no—
118C2×C4×D4—93no—
119#395—31no—
120#394—93no—
121#393—31no—
122#392—93no—
123C4×C22⋊C4—93no—
124C24×C3—31no—
125C23×D3—33no—
126C23×D3—33no—
127C23×D3—63no—
128C23×D3C22×D311yescharacteristic
129C2×C3×Q8—212no—
130C2×C3×Q8—26no—
131C2×C3×Q8—26no—
132C22×C3×C4—23no—
133C22×C3×C4—33no—
134C22×C3×C4—33no—
135C22×C3×C4—31no—
136C22×C3×C4C22×D313yes—
137C22×C3×C4C22×D313yes—
138C2×#67—66no—
139C2×#67—66no—
140C2×#67—66no—
141C22×Dic3—31no—
142C22×Dic3—31no—
143C22×Dic3—61no—
144C22×Dic3—66no—
145C22×Dic3C22×D313yes—
146C22×Dic3C22×D311yescharacteristic
147C22×Dic3C22×D313yes—
148#239—632no—
149#239—632no—
150#239—664no—
151C2×D12—612no—
152C2×D12—612no—
153C2×D12—624no—
154C2×D12—612no—
155C2×D12—66no—
156C2×D12—66no—
157C2×D12—66no—
158C2×D12—66no—
159C2×D12—66no—
160C2×C4×D3—312no—
161C2×C4×D3—312no—
162C2×C4×D3—612no—
163C2×C4×D3—612no—
164C2×C4×D3—612no—
165C2×C4×D3—624no—
166C2×C4×D3—66no—
167C3×C4⋊C4—66no—
168C3×C22⋊C4—66no—
169(C2×D3)⋊C4—26no—
170(C2×D3)⋊C4—612no—
171(C2×D3)⋊C4—66no—
172(C2×D3)⋊C4—66no—
173(C2×D3)⋊C4—612no—
174Dic3⋊C4—26no—
175C4×Dic3—66no—
176C4×Dic3—66no—
177C22×C32C22×D413yes—
178C22×C32C2×#4111yescharacteristic
179C22×C32C2×#4113yes—
180C2×C3⋊D3—26no—
181C2×C3⋊D3—26no—
182C2×C3⋊D3—26no—
183C2×C3⋊D3—46no—
184C2×C3⋊D3—46no—
185C2×C3×D3—26no—
186C2×C3×D3—41no—
187C2×C3×D3—43no—
188C32×C4—26no—
189C3⋊Dic3#12014yes—
190C3×Dic3—26no—
191C3×Dic3—41no—
192C3×Dic3—43no—
193C2×#41—1812no—
194C22×Q8D3211yescharacteristic
195C22×D4—93no—
196C22×D4—93no—
197C23×C4—91no—
198C23×C4—93no—
199C4×D4—1812no—
200C2×C4⋊C4—33no—
201C2×C22⋊C4—33no—
202C2×C22⋊C4—93no—
203C2×C22⋊C4—96no—
204C2×C42—93no—
205#95—33no—
206#95—33no—
207C23×C3—21no—
208C23×C3—32no—
209C23×C3—63no—
210C23×C3—63no—
211C23×C3C23×D311yescharacteristic
212C23×C3C23×D311yescharacteristic
213C22×D3—23no—
214C22×D3—23no—
215C22×D3—36no—
216C22×D3—36no—
217C22×D3—66no—
218C22×D3—612no—
219C22×D3—66no—
220C22×D3—66no—
221C22×D3—66no—
222C22×D3—63no—
223C22×D3—63no—
224C22×D3—66no—
225C22×D3—63no—
226C22×D3—63no—
227C22×D3—126no—
228C22×D3—123no—
229C22×D3—123no—
230C22×D3#23912yescomplemented
231C3×Q8—216no—
232C3×Q8—28no—
233C3×Q8—28no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—26no—
237C2×C3×C4—26no—
238C2×C3×C4—36no—
239C2×C3×C4—63no—
240C2×C3×C4—63no—
241C2×C3×C4—63no—
242C2×C3×C4—63no—
243C2×C3×C4—66no—
244C2×C3×C4D3×D416yes—
245C2×C3×C4D3×D416yes—
246#67—624no—
247C2×Dic3—23no—
248C2×Dic3—23no—
249C2×Dic3—23no—
250C2×Dic3—23no—
251C2×Dic3—26no—
252C2×Dic3—36no—
253C2×Dic3—36no—
254C2×Dic3—66no—
255C2×Dic3—124no—
256C2×Dic3#23916yescomplemented
257D12—648no—
258D12—624no—
259D12—624no—
260D12—624no—
261C4×D3—624no—
262C4×D3—624no—
263C4×D3—648no—
264C4×D3—124no—
265C2×C32C2×#12011yescharacteristic
266C2×C32#52513yes—
267C2×C32#53213yes—
268C3⋊D3—46no—
269C3×D3—42no—
270C24—31no—
271C24—93no—
272#41—1832no—
273C2×Q8—26no—
274C2×D4—1812no—
275C2×D4—186no—
276C2×D4—186no—
277C2×D4—186no—
278C2×D4—186no—
279C22×C4—33no—
280C22×C4—33no—
281C22×C4—31no—
282C22×C4—91no—
283C22×C4—912no—
284C22×C4—1812no—
285C22×C4—186no—
286C22×C4C2×D3213yes—
287C4⋊C4—66no—
288C22⋊C4—66no—
289C22⋊C4—1812no—
290C22⋊C4—186no—
291C42—186no—
292C22×C3—23no—
293C22×C3—21no—
294C22×C3—23no—
295C22×C3—66no—
296C22×C3—123no—
297C22×C3—121no—
298C22×C3C2×D3×D413yes—
299C22×C3C2×D3×D413yes—
300C22×C3C2×#23913yes—
301C22×C3C2×#23911yescharacteristic
302C22×C3D3×#4113yes—
303C22×C3D3×#4111yescharacteristic
304C2×D3—26no—
305C2×D3—43no—
306C2×D3—41no—
307C2×D3—612no—
308C2×D3—66no—
309C2×D3—66no—
310C2×D3—66no—
311C2×D3—66no—
312C2×D3—66no—
313C2×D3—66no—
314C2×D3—126no—
315C2×D3—126no—
316C2×D3—1212no—
317C2×D3—1212no—
318C2×D3—126no—
319C2×D3—126no—
320C2×D3—126no—
321C2×D3—126no—
322C3×C4—212no—
323C3×C4—26no—
324C3×C4—26no—
325C3×C4—66no—
326C3×C4—123no—
327C3×C4—121no—
328Dic3—26no—
329Dic3—34no—
330Dic3—34no—
331Dic3—43no—
332Dic3—41no—
333Dic3—64no—
334C32#238811yescharacteristic, complemented, Sylow
335C23—32no—
336C23—63no—
337C23—63no—
338C23—96no—
339C23—186no—
340C23—183no—
341C23—183no—
342C23—186no—
343C23C22×D3211yescharacteristic
344Q8—28no—
345D4—1824no—
346D4—1824no—
347C2×C4—26no—
348C2×C4—36no—
349C2×C4—63no—
350C2×C4—63no—
351C2×C4—66no—
352C2×C4—63no—
353C2×C4—63no—
354C2×C4—96no—
355C2×C4—1824no—
356C2×C4—364no—
357C2×C4#381116yes—
358C2×C3—23no—
359C2×C3—23no—
360C2×C3—21no—
361C2×C3—122no—
362C2×C3D3×#12011yescharacteristic
363C2×C3D3×#12011yescharacteristic
364C2×C3#562713yes—
365C2×C3#562713yes—
366C2×C3#566813yes—
367C2×C3#572513yes—
368D3—42no—
369D3—1212no—
370D3—126no—
371D3—126no—
372C22—66no—
373C22—123no—
374C22—121no—
375C22—186no—
376C22—186no—
377C22—186no—
378C22—366no—
379C22—366no—
380C22D3×#23913yes—
381C22D3×#23911yescharacteristic
382C22C2×#381113yes—
383C4—26no—
384C4—66no—
385C4—94no—
386C4—123no—
387C4—121no—
388C3—21no—
389C3#7677711yescharacteristic, complemented
390C3#7716911yescharacteristic, complemented
391C2—122no—
392C2—366no—
393C2#1059536313yes—
394C2#1059543013yes—
395C2#1059552711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC22×C32×Q8C22
hypercenterC22×Q8D32
hypocenterC32#2388
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3832C2semidirect
2. C2×#63881C2semidirect
3. C3×#5231C2semidirect
4. C3×#5230C2semidirect
5. #10591395C2semidirect
6. #10591321C2semidirect
7. #10591260C2semidirect
31. #5231D3semidirect
32. #5230D3semidirect
48. C2×C4×C3⋊D3D4semidirect
102. C22×C3×D3#41semidirect
114. C2×C3×Dic3#41semidirect
230. C22×D3#239semidirect
256. C2×Dic3#239semidirect
334. C32#2388semidirect
389. C3#76777semidirect
390. C3#77169semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
C2×#638812
C3×#52312
C3×#52302
#105913952
#105913212
#105912602
#771693
#767773