Finite Group Databasebeta

← #11722507

Subgroups — #11722507

Order 864.

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

normal subgroups only

This group has 635 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
1D4×#272C211yesmaximal, characteristic, complemented
2#3830⋊C3C211yesmaximal, characteristic, complemented
3D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#88677C212yesmaximal, complemented
5#88676C212yesmaximal, complemented
6C2×#6200C212yesmaximal, complemented
7#88671C212yesmaximal, complemented
8#88654C211yesmaximal, characteristic, complemented
9#88652C211yesmaximal, characteristic, complemented
10#88650C211yesmaximal, characteristic, complemented
11#88649C211yesmaximal, characteristic, complemented
12#64314—31nomaximal
13#64313—31nomaximal
14C22×#272C2212yescomplemented
15C22×(C3⋊D3)⋊C3C2212yescomplemented
16C2×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
17D4×H(3,𝔽3)C2211yescharacteristic, complemented
18#6235C2212yescomplemented
19C4⋊#272C2211yescharacteristic, complemented
20C4×#272C2211yescharacteristic, complemented
21#638⋊C3C2212yescomplemented
22#638⋊C3C2212yescomplemented
23C2×(C3⋊Dic3)⋊C3C2212yescomplemented
24C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
25C4×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
26C4×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
27#634⋊C3C2211yescharacteristic, complemented
28#6200C2212yescomplemented
29#6200C2211yescharacteristic, complemented
30#6200C2211yescharacteristic, complemented
31#6200C2212yescomplemented
32#6200C2212yescomplemented
33#6199C2212yescomplemented
34#6198C2211yescharacteristic, complemented
35((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
36#6196C2211yescharacteristic, complemented
37#3830D311yescharacteristic, complemented
38D4×C3⋊D3D311yescharacteristic, complemented
39C3×#237—31no—
40C3×D3×D4—31no—
41C3×D3×D4—31no—
42C3×D3×D4—31no—
43C3×D3×D4—61no—
44#3811—32no—
45D3×#67—32no—
46D3×#67—32no—
47D3×#67—32no—
48C2×#626—32no—
49C2×#625—32no—
50#3805—32no—
51#3804—32no—
52#3802—31no—
53D3×D12—31no—
54#3799—31no—
55#3798—31no—
56#3798—31no—
57#3796—31no—
58#3795—31no—
59#3795—31no—
60C22×H(3,𝔽3)C2312yes—
61C2×#272—42no—
62C2×#272C2312yes—
63C2×#272C2311yescharacteristic
64C2×(C3⋊D3)⋊C3—42no—
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(C3⋊Dic3)⋊C3C2311yescharacteristic
72(C3⋊Dic3)⋊C3C2311yescharacteristic
73(C3⋊Dic3)⋊C3C2312yes—
74#1009—91no—
75#1004—91no—
76#1004—91no—
77C22×C3⋊D3C2×D312yescomplemented
78C22×C3×D3—32no—
79C22×C3×D3—32no—
80C22×C3×D3—32no—
81C22×C3×D3—62no—
82C2×D32—32no—
83C2×D32—32no—
84C32×D4—21no—
85C32×D4C2×D311yescharacteristic, complemented
86C32×D4C2×D311yescharacteristic, complemented
87#638C2×D312yescomplemented
88#638C2×D312yescomplemented
89C2×C3⋊Dic3C2×D312yescomplemented
90C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
91C4×C3⋊D3C2×D311yescharacteristic, complemented
92C4×C3⋊D3C2×D311yescharacteristic, complemented
93#634C2×D311yescharacteristic, complemented
94C3×#67—32no—
95C3×#67—32no—
96C3×#67—32no—
97C3×#67—32no—
98C3×#67—62no—
99C2×C3×Dic3—32no—
100C3×D12—31no—
101C3×D12—31no—
102C3×D12—31no—
103C3×D12—61no—
104C3×C4×D3—31no—
105C3×C4×D3—31no—
106C3×C4×D3—31no—
107C3×C4×D3—31no—
108C3×C4×D3—61no—
109C3×Dic6—31no—
110#627—31no—
111#627—31no—
112#626—31no—
113#626—31no—
114#626—32no—
115#626—31no—
116#626—31no—
117#626—32no—
118#626—32no—
119#625—32no—
120#625—32no—
121#625—31no—
122#625—31no—
123#625—32no—
124#624—32no—
125D3×Dic3—32no—
126D3×Dic3—32no—
127D3×Dic3—32no—
128C2×H(3,𝔽3)—22no—
129C2×H(3,𝔽3)C2411yescharacteristic
130#272—21no—
131#272—22no—
132(C3⋊D3)⋊C3—21no—
133(C3⋊D3)⋊C3—22no—
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—91no—
146#237—92no—
147#237—92no—
148#237—91no—
149#237—181no—
150D3×D4—31no—
151D3×D4—31no—
152D3×D4—31no—
153D3×D4—61no—
154D3×D4—91no—
155D3×D4—92no—
156D3×D4—92no—
157D3×D4—92no—
158D3×D4—92no—
159D3×D4—91no—
160D3×D4—181no—
161#235—91no—
162#235—91no—
163#235—91no—
164#235—91no—
165#235—92no—
166#235—91no—
167C2×D12—91no—
168C2×D12—92no—
169C22×C32—22no—
170C22×C32C22×D312yes—
171C22×C32C22×D312yes—
172C2×C3⋊D3—42no—
173C2×C3⋊D3C22×D312yes—
174C2×C3⋊D3C22×D311yescharacteristic
175C2×C3⋊D3C22×D311yescharacteristic
176C2×C3×D3—32no—
177C2×C3×D3—31no—
178C2×C3×D3—31no—
179C2×C3×D3—32no—
180C2×C3×D3—32no—
181C2×C3×D3—31no—
182C2×C3×D3—31no—
183C2×C3×D3—61no—
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187C2×C3×D3—122no—
188C2×C3×D3—124no—
189D32—122no—
190D32—122no—
191C32×C4—21no—
192C32×C4C22×D311yescharacteristic
193C32×C4C22×D311yescharacteristic
194C3⋊Dic3C22×D312yes—
195C3⋊Dic3C22×D311yescharacteristic
196C3⋊Dic3C22×D311yescharacteristic
197C3×Dic3—31no—
198C3×Dic3—32no—
199C3×Dic3—31no—
200C3×Dic3—31no—
201C3×Dic3—31no—
202C3×Dic3—61no—
203H(5,𝔽2)—271noSylow
204H(3,𝔽3)H(5,𝔽2)11yescharacteristic, complemented, Sylow
205C23×C3—92no—
206C23×C3—92no—
207C22×D3—32no—
208C22×D3—32no—
209C22×D3—32no—
210C22×D3—62no—
211C22×D3—92no—
212C22×D3—92no—
213C22×D3—92no—
214C22×D3—182no—
215C3×Q8—91no—
216C3×D4—61no—
217C3×D4—61no—
218C3×D4—61no—
219C3×D4—91no—
220C3×D4—92no—
221C3×D4—92no—
222C3×D4—92no—
223C3×D4—91no—
224C3×D4D3211yescharacteristic
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—92no—
228C2×C3×C4—91no—
229#67—32no—
230#67—32no—
231#67—32no—
232#67—32no—
233#67—62no—
234#67—92no—
235#67—91no—
236#67—91no—
237#67—92no—
238#67—92no—
239#67—91no—
240#67—91no—
241#67—91no—
242#67—91no—
243#67—92no—
244#67—92no—
245#67—92no—
246#67—182no—
247#67—182no—
248C2×Dic3—32no—
249C2×Dic3—92no—
250C2×Dic3—92no—
251C2×Dic3—182no—
252D12—31no—
253D12—31no—
254D12—31no—
255D12—61no—
256D12—91no—
257D12—91no—
258D12—92no—
259D12—92no—
260D12—92no—
261D12—91no—
262D12—181no—
263C4×D3—31no—
264C4×D3—31no—
265C4×D3—31no—
266C4×D3—31no—
267C4×D3—61no—
268C4×D3—92no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—92no—
272C4×D3—181no—
273C4×D3—181no—
274Dic6—31no—
275Dic6—91no—
276Dic6—91no—
277Dic6—91no—
278Dic6—181no—
279C2×C32—22no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—42no—
283C2×C32C23×D311yescharacteristic
284C2×C32C23×D311yescharacteristic
285C3⋊D3—21no—
286C3⋊D3—22no—
287C3⋊D3—21no—
288C3×D3—62no—
289C3×D3—61no—
290C3×D3—62no—
291C3×D3—61no—
292C3×D3—62no—
293C3×D3—61no—
294C3×D3—62no—
295C3×D3—61no—
296C3×D3—122no—
297#41—91no—
298#41—271no—
299#41—272no—
300#41—271no—
301#41—271no—
302C2×D4—91no—
303C2×D4—91no—
304C2×D4—271no—
305C2×D4—272no—
306C2×D4—272no—
307C2×D4—272no—
308C22×C3—62no—
309C22×C3—62no—
310C22×C3—62no—
311C22×C3—92no—
312C22×C3—91no—
313C22×C3—91no—
314C22×C3—92no—
315C22×C3—91no—
316C22×C3—362no—
317C22×C3—362no—
318C22×C3C2×D3212yes—
319C2×D3—31no—
320C2×D3—31no—
321C2×D3—32no—
322C2×D3—32no—
323C2×D3—31no—
324C2×D3—31no—
325C2×D3—32no—
326C2×D3—61no—
327C2×D3—62no—
328C2×D3—122no—
329C2×D3—122no—
330C2×D3—122no—
331C2×D3—124no—
332C2×D3—181no—
333C2×D3—182no—
334C2×D3—181no—
335C2×D3—362no—
336C2×D3—362no—
337C2×D3—362no—
338C2×D3—362no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—91no—
343C3×C4—91no—
344C3×C4—92no—
345C3×C4—91no—
346C3×C4C2×D3211yescharacteristic
347Dic3—31no—
348Dic3—31no—
349Dic3—32no—
350Dic3—31no—
351Dic3—31no—
352Dic3—61no—
353Dic3—181no—
354Dic3—181no—
355Dic3—182no—
356C32—21no—
357C32#100411yescharacteristic, complemented
358C32#100911yescharacteristic, complemented
359C23—92no—
360C23—92no—
361C23—272no—
362Q8—91no—
363Q8—271no—
364D4—92no—
365D4—92no—
366D4—92no—
367D4—91no—
368D4—91no—
369D4—271no—
370D4—272no—
371D4—271no—
372D4—271no—
373D4—272no—
374D4—272no—
375D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
376C2×C4—92no—
377C2×C4—91no—
378C2×C4—91no—
379C2×C4—91no—
380C2×C4—272no—
381C2×C4—272no—
382C2×C3—22no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—122no—
387C2×C3—122no—
388C2×C3—122no—
389C2×C3—181no—
390C2×C3—182no—
391C2×C3—181no—
392C2×C3—182no—
393C2×C3—181no—
394C2×C3C22×D3211yescharacteristic
395D3—61no—
396D3—61no—
397D3—62no—
398D3—62no—
399D3—62no—
400D3—61no—
401D3—62no—
402D3—61no—
403D3—122no—
404D3—181no—
405D3—362no—
406D3—361no—
407C22—91no—
408C22—91no—
409C22—91no—
410C22—92no—
411C22—92no—
412C22—362no—
413C22—362no—
414C22—1082no—
415C22C2×((C3⋊D3)⋊C3)⋊C212yes—
416C4—91no—
417C4—91no—
418C4—91no—
419C4—92no—
420C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C3—61no—
422C3—61no—
423C3—61no—
424C3#6431411yescharacteristic
425C2—22no—
426C2—181no—
427C2—181no—
428C2—182no—
429C2—182no—
430C2—181no—
431C2C22×((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. D4×#272C2semidirect
2. #3830⋊C3C2semidirect
3. D4×(C3⋊D3)⋊C3C2semidirect
4. #88677C2semidirect
5. #88676C2semidirect
6. C2×#6200C2semidirect
7. #88671C2semidirect
8. #88654C2semidirect
9. #88652C2semidirect
10. #88650C2semidirect
11. #88649C2semidirect
14. C22×#272C22semidirect
15. C22×(C3⋊D3)⋊C3C22semidirect
16. C2×((C3⋊D3)⋊C3)⋊C2C22semidirect
17. D4×H(3,𝔽3)C22semidirect
18. #6235C22semidirect
19. C4⋊#272C22semidirect
20. C4×#272C22semidirect
21. #638⋊C3C22semidirect
22. #638⋊C3C22semidirect
23. C2×(C3⋊Dic3)⋊C3C22semidirect
24. C4⋊((C3⋊D3)⋊C3)C22semidirect
25. C4×(C3⋊D3)⋊C3C22semidirect
26. C4×(C3⋊D3)⋊C3C22semidirect
27. #634⋊C3C22semidirect
28. #6200C22semidirect
29. #6200C22semidirect
30. #6200C22semidirect
31. #6200C22semidirect
32. #6200C22semidirect
33. #6199C22semidirect
34. #6198C22semidirect
35. ((C3⋊D3)⋊C3)⋊C4C22semidirect
36. #6196C22semidirect
37. #3830D3semidirect
38. D4×C3⋊D3D3semidirect
77. C22×C3⋊D3C2×D3semidirect
85. C32×D4C2×D3semidirect
86. C32×D4C2×D3semidirect
87. #638C2×D3semidirect
88. #638C2×D3semidirect
89. C2×C3⋊Dic3C2×D3semidirect
90. C4⋊(C3⋊D3)C2×D3semidirect
91. C4×C3⋊D3C2×D3semidirect
92. C4×C3⋊D3C2×D3semidirect
93. #634C2×D3semidirect
204. H(3,𝔽3)H(5,𝔽2)semidirect
357. C32#1004semidirect
358. C32#1009semidirect
375. 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
D4×#2722
#3830⋊C32
D4×(C3⋊D3)⋊C32
#886772
#886762
C2×#62002
#886712
#886542
#886522
#886502
#886492
#643143
#643133