Finite Group Databasebeta

← #11722506

Subgroups — #11722506

Order 864.

621 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#88744C211yesmaximal, characteristic, complemented
2D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#88676C212yesmaximal, complemented
5#88676C212yesmaximal, complemented
6C2×#6198C212yesmaximal, complemented
7#88672C212yesmaximal, complemented
8#88655C211yesmaximal, characteristic, complemented
9#88650C211yesmaximal, characteristic, complemented
10#88648C211yesmaximal, characteristic, complemented
11#88648C211yesmaximal, characteristic, complemented
12#64314—31nomaximal
13#64314—31nomaximal
14C22×(C3⋊D3)⋊C3C2212yescomplemented
15C22×(C3⋊D3)⋊C3C2212yescomplemented
16C2×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
17D4×H(3,𝔽3)C2211yescharacteristic, complemented
18#6235C2212yescomplemented
19C2×#1084C2212yescomplemented
20C4×#272C2211yescharacteristic, complemented
21#6229C2211yescharacteristic, complemented
22#638⋊C3C2212yescomplemented
23#638⋊C3C2212yescomplemented
24C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
25C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
26C4×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
27C4×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
28#6200C2211yescharacteristic, complemented
29#6200C2211yescharacteristic, complemented
30#6198C2212yescomplemented
31#6198C2212yescomplemented
32#6198C2211yescharacteristic, complemented
33#6198C2212yescomplemented
34((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
35((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
36#6196C2211yescharacteristic, complemented
37D4×C3⋊D3D311yescharacteristic, complemented
38D4×C3⋊D3D311yescharacteristic, complemented
39C3×#237—31no—
40C3×#237—31no—
41C3×#237—61no—
42C3×D3×D4—31no—
43C3×D3×D4—31no—
44#3811—32no—
45#3811—32no—
46D3×#67—32no—
47D3×#67—32no—
48C2×#626—32no—
49C2×#626—32no—
50#3804—32no—
51#3804—32no—
52D3×D12—31no—
53D3×D12—31no—
54#3799—31no—
55#3799—31no—
56#3798—31no—
57#3798—31no—
58#3796—31no—
59#3796—31no—
60C22×H(3,𝔽3)C2312yes—
61C2×#272C2311yescharacteristic
62C2×(C3⋊D3)⋊C3—42no—
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3C2312yes—
65C2×(C3⋊D3)⋊C3C2312yes—
66C2×(C3⋊D3)⋊C3C2311yescharacteristic
67C2×(C3⋊D3)⋊C3C2311yescharacteristic
68((C3⋊D3)⋊C3)⋊C2—42no—
69C4×H(3,𝔽3)C2311yescharacteristic
70#1084C2311yescharacteristic
71#1084C2312yes—
72(C3⋊Dic3)⋊C3C2311yescharacteristic
73(C3⋊Dic3)⋊C3C2311yescharacteristic
74#1009—91no—
75#1004—91no—
76#1004—91no—
77C22×C3⋊D3C2×D312yescomplemented
78C22×C3⋊D3C2×D312yescomplemented
79C22×C3×D3—32no—
80C22×C3×D3—32no—
81C2×D32—32no—
82C2×D32—32no—
83C32×D4—21no—
84C32×D4C2×D311yescharacteristic, complemented
85C32×D4C2×D311yescharacteristic, complemented
86#638C2×D312yescomplemented
87#638C2×D312yescomplemented
88C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
89C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
90C4×C3⋊D3C2×D311yescharacteristic, complemented
91C4×C3⋊D3C2×D311yescharacteristic, complemented
92C3×#67—32no—
93C3×#67—32no—
94C3×#67—32no—
95C3×#67—32no—
96C3×#67—62no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—62no—
100C3×D12—31no—
101C3×D12—31no—
102C3×C4×D3—31no—
103C3×C4×D3—31no—
104C3×C4×D3—31no—
105C3×C4×D3—31no—
106C3×C4×D3—61no—
107C3×Dic6—31no—
108C3×Dic6—31no—
109C3×Dic6—61no—
110#627—31no—
111#627—31no—
112#626—32no—
113#626—32no—
114#626—31no—
115#626—31no—
116#626—32no—
117#626—32no—
118#626—32no—
119#626—32no—
120#626—31no—
121#626—31no—
122#625—31no—
123#625—31no—
124#624—32no—
125#624—32no—
126D3×Dic3—32no—
127D3×Dic3—32no—
128C2×H(3,𝔽3)—22no—
129C2×H(3,𝔽3)C2411yescharacteristic
130#272—21no—
131(C3⋊D3)⋊C3—22no—
132(C3⋊D3)⋊C3—22no—
133(C3⋊D3)⋊C3—21no—
134(C3⋊D3)⋊C3—21no—
135C3×#41—91no—
136C2×C3×D4—91no—
137C2×C3×D4—91no—
138C2×#67—92no—
139C2×#67—92no—
140C2×#67—92no—
141C2×#67—92no—
142#239—91no—
143#239—91no—
144#237—31no—
145#237—31no—
146#237—61no—
147#237—91no—
148#237—91no—
149#237—92no—
150#237—92no—
151D3×D4—31no—
152D3×D4—31no—
153D3×D4—91no—
154D3×D4—91no—
155D3×D4—91no—
156D3×D4—92no—
157D3×D4—92no—
158D3×D4—92no—
159D3×D4—92no—
160D3×D4—181no—
161#235—91no—
162#235—91no—
163#235—91no—
164#235—91no—
165#235—91no—
166#235—92no—
167C2×D12—91no—
168C2×D12—92no—
169C22×C32—22no—
170C22×C32C22×D312yes—
171C22×C32C22×D312yes—
172C2×C3⋊D3—42no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3C22×D312yes—
175C2×C3⋊D3C22×D311yescharacteristic
176C2×C3⋊D3C22×D312yes—
177C2×C3⋊D3C22×D311yescharacteristic
178C2×C3×D3—32no—
179C2×C3×D3—32no—
180C2×C3×D3—31no—
181C2×C3×D3—31no—
182C2×C3×D3—31no—
183C2×C3×D3—31no—
184C2×C3×D3—61no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187D32—122no—
188D32—122no—
189C32×C4—21no—
190C32×C4C22×D311yescharacteristic
191C32×C4C22×D311yescharacteristic
192C3⋊Dic3C22×D311yescharacteristic
193C3⋊Dic3C22×D311yescharacteristic
194C3×Dic3—31no—
195C3×Dic3—31no—
196C3×Dic3—32no—
197C3×Dic3—32no—
198C3×Dic3—31no—
199C3×Dic3—31no—
200C3×Dic3—62no—
201C3×Dic3—61no—
202H(5,𝔽2)—271noSylow
203H(3,𝔽3)H(5,𝔽2)11yescharacteristic, complemented, Sylow
204C23×C3—92no—
205C23×C3—92no—
206C22×D3—32no—
207C22×D3—32no—
208C22×D3—92no—
209C22×D3—92no—
210C22×D3—92no—
211C22×D3—92no—
212C22×D3—182no—
213C3×Q8—91no—
214C3×D4—31no—
215C3×D4—61no—
216C3×D4—61no—
217C3×D4—91no—
218C3×D4—91no—
219C3×D4—92no—
220C3×D4—92no—
221C3×D4—92no—
222C3×D4D3211yescharacteristic
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—92no—
226C2×C3×C4—91no—
227#67—32no—
228#67—32no—
229#67—32no—
230#67—32no—
231#67—62no—
232#67—92no—
233#67—91no—
234#67—91no—
235#67—92no—
236#67—91no—
237#67—92no—
238#67—92no—
239#67—92no—
240#67—91no—
241#67—92no—
242#67—92no—
243#67—91no—
244#67—91no—
245#67—182no—
246C2×Dic3—32no—
247C2×Dic3—32no—
248C2×Dic3—62no—
249C2×Dic3—92no—
250C2×Dic3—92no—
251D12—31no—
252D12—31no—
253D12—91no—
254D12—91no—
255D12—92no—
256D12—92no—
257D12—92no—
258D12—91no—
259D12—91no—
260D12—181no—
261C4×D3—31no—
262C4×D3—31no—
263C4×D3—31no—
264C4×D3—31no—
265C4×D3—61no—
266C4×D3—91no—
267C4×D3—92no—
268C4×D3—92no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—181no—
272Dic6—31no—
273Dic6—31no—
274Dic6—61no—
275Dic6—91no—
276Dic6—91no—
277Dic6—91no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—21no—
281C2×C32—42no—
282C2×C32C23×D311yescharacteristic
283C2×C32C23×D311yescharacteristic
284C3⋊D3—22no—
285C3⋊D3—21no—
286C3⋊D3—21no—
287C3⋊D3—22no—
288C3×D3—62no—
289C3×D3—62no—
290C3×D3—61no—
291C3×D3—61no—
292C3×D3—61no—
293C3×D3—61no—
294C3×D3—121no—
295#41—91no—
296#41—271no—
297#41—272no—
298#41—271no—
299#41—271no—
300C2×D4—91no—
301C2×D4—91no—
302C2×D4—271no—
303C2×D4—272no—
304C2×D4—272no—
305C2×D4—272no—
306C22×C3—32no—
307C22×C3—62no—
308C22×C3—62no—
309C22×C3—92no—
310C22×C3—91no—
311C22×C3—92no—
312C22×C3—91no—
313C22×C3—91no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3C2×D3212yes—
317C2×D3—31no—
318C2×D3—31no—
319C2×D3—31no—
320C2×D3—31no—
321C2×D3—32no—
322C2×D3—32no—
323C2×D3—61no—
324C2×D3—91no—
325C2×D3—92no—
326C2×D3—122no—
327C2×D3—122no—
328C2×D3—181no—
329C2×D3—182no—
330C2×D3—362no—
331C2×D3—362no—
332C2×D3—362no—
333C2×D3—362no—
334C2×D3—362no—
335C3×C4—31no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—91no—
339C3×C4—91no—
340C3×C4—91no—
341C3×C4—92no—
342C3×C4C2×D3211yescharacteristic
343Dic3—32no—
344Dic3—31no—
345Dic3—32no—
346Dic3—31no—
347Dic3—31no—
348Dic3—31no—
349Dic3—62no—
350Dic3—61no—
351Dic3—91no—
352Dic3—181no—
353C32—21no—
354C32#100411yescharacteristic, complemented
355C32#100411yescharacteristic, complemented
356C23—92no—
357C23—92no—
358C23—272no—
359Q8—91no—
360Q8—271no—
361D4—92no—
362D4—92no—
363D4—92no—
364D4—91no—
365D4—91no—
366D4—271no—
367D4—272no—
368D4—271no—
369D4—271no—
370D4—272no—
371D4—272no—
372D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
373C2×C4—92no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—91no—
377C2×C4—272no—
378C2×C4—272no—
379C2×C3—22no—
380C2×C3—31no—
381C2×C3—62no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—122no—
385C2×C3—122no—
386C2×C3—182no—
387C2×C3—181no—
388C2×C3—182no—
389C2×C3—181no—
390C2×C3—181no—
391C2×C3C22×D3211yescharacteristic
392D3—61no—
393D3—61no—
394D3—61no—
395D3—61no—
396D3—62no—
397D3—62no—
398D3—121no—
399D3—181no—
400D3—182no—
401D3—181no—
402D3—362no—
403C22—91no—
404C22—91no—
405C22—91no—
406C22—92no—
407C22—92no—
408C22—362no—
409C22—362no—
410C22—1082no—
411C22C2×((C3⋊D3)⋊C3)⋊C212yes—
412C4—91no—
413C4—91no—
414C4—91no—
415C4—92no—
416C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C3—31no—
418C3—61no—
419C3—61no—
420C3#6431311yescharacteristic
421C2—22no—
422C2—181no—
423C2—181no—
424C2—182no—
425C2—182no—
426C2—181no—
427C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
Frattini, socleC2×C3C22×D32
FittingD4×H(3,𝔽3)C22
hypercenterD4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #88744C2semidirect
2. D4×(C3⋊D3)⋊C3C2semidirect
3. D4×(C3⋊D3)⋊C3C2semidirect
4. #88676C2semidirect
5. #88676C2semidirect
6. C2×#6198C2semidirect
7. #88672C2semidirect
8. #88655C2semidirect
9. #88650C2semidirect
10. #88648C2semidirect
11. #88648C2semidirect
14. C22×(C3⋊D3)⋊C3C22semidirect
15. C22×(C3⋊D3)⋊C3C22semidirect
16. C2×((C3⋊D3)⋊C3)⋊C2C22semidirect
17. D4×H(3,𝔽3)C22semidirect
18. #6235C22semidirect
19. C2×#1084C22semidirect
20. C4×#272C22semidirect
21. #6229C22semidirect
22. #638⋊C3C22semidirect
23. #638⋊C3C22semidirect
24. C4⋊((C3⋊D3)⋊C3)C22semidirect
25. C4⋊((C3⋊D3)⋊C3)C22semidirect
26. C4×(C3⋊D3)⋊C3C22semidirect
27. C4×(C3⋊D3)⋊C3C22semidirect
28. #6200C22semidirect
29. #6200C22semidirect
30. #6198C22semidirect
31. #6198C22semidirect
32. #6198C22semidirect
33. #6198C22semidirect
34. ((C3⋊D3)⋊C3)⋊C4C22semidirect
35. ((C3⋊D3)⋊C3)⋊C4C22semidirect
36. #6196C22semidirect
37. D4×C3⋊D3D3semidirect
38. D4×C3⋊D3D3semidirect
77. C22×C3⋊D3C2×D3semidirect
78. C22×C3⋊D3C2×D3semidirect
84. C32×D4C2×D3semidirect
85. C32×D4C2×D3semidirect
86. #638C2×D3semidirect
87. #638C2×D3semidirect
88. C4⋊(C3⋊D3)C2×D3semidirect
89. C4⋊(C3⋊D3)C2×D3semidirect
90. C4×C3⋊D3C2×D3semidirect
91. C4×C3⋊D3C2×D3semidirect
203. H(3,𝔽3)H(5,𝔽2)semidirect
354. C32#1004semidirect
355. C32#1004semidirect
372. D4((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
#887442
D4×(C3⋊D3)⋊C32
#886762
C2×#61982
#886722
#886552
#886502
#886482
#643143