Finite Group Databasebeta

← #11722491

Subgroups — #11722491

Order 864.

628 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 116 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×C4×#272C211yesmaximal, characteristic, complemented
2#3828⋊C3C211yesmaximal, characteristic, complemented
3#3828⋊C3C211yesmaximal, characteristic, complemented
4#88677C212yesmaximal, complemented
5#88672C212yesmaximal, complemented
6#88655C211yesmaximal, characteristic, complemented
7C4×((C3⋊D3)⋊C3)⋊C2C212yesmaximal, complemented
8#88651C211yesmaximal, characteristic, complemented
9#88650C212yesmaximal, complemented
10#88649C211yesmaximal, characteristic, complemented
11#88649C211yesmaximal, characteristic, complemented
12D3×#235—31nomaximal
13D3×#235—31nomaximal
14C22×#272C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C2C2212yes—
16C2×C4×H(3,𝔽3)C2211yescharacteristic, complemented
17C2×#1084C2211yescharacteristic, complemented
18C4×#272C2211yescharacteristic
19C4×#272C2211yescharacteristic
20C4×#272C2211yescharacteristic, complemented
21C4×#272C2211yescharacteristic, complemented
22#638⋊C3C2212yescomplemented
23#638⋊C3C2212yescomplemented
24C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
25C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
26C4×(C3⋊D3)⋊C3C2212yescomplemented
27C4×(C3⋊D3)⋊C3C2212yescomplemented
28#634⋊C3C2211yescharacteristic, complemented
29#634⋊C3C2211yescharacteristic, complemented
30#6200C2212yescomplemented
31#6200C2212yescomplemented
32#6199C2212yes—
33#6198C2212yescomplemented
34((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
35((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
36#6196C2212yescomplemented
37#3828D311yescharacteristic, complemented
38#3828D311yescharacteristic, complemented
39C3×#235—31no—
40C3×#235—31no—
41C2×C3×C4×D3—31no—
42C2×C3×C4×D3—31no—
43C2×C3×C4×D3—61no—
44D3×#67—32no—
45D3×#67—32no—
46#3804—32no—
47#3804—32no—
48D3×D12—31no—
49D3×D12—31no—
50C4×D32—32no—
51C4×D32—32no—
52#3799—31no—
53#3799—31no—
54#3798—32no—
55#3798—32no—
56#3795—31no—
57#3795—31no—
58D3×Dic6—31no—
59D3×Dic6—31no—
60C22×H(3,𝔽3)C2311yescharacteristic
61C2×#272—22no—
62C2×#272C2311yescharacteristic
63C2×#272C2311yescharacteristic
64C2×(C3⋊D3)⋊C3C2312yes—
65C2×(C3⋊D3)⋊C3C2312yes—
66((C3⋊D3)⋊C3)⋊C2—24no—
67C4×H(3,𝔽3)C2311yescharacteristic
68C4×H(3,𝔽3)C2311yescharacteristic
69#1084C2311yescharacteristic
70#1084C2311yescharacteristic
71(C3⋊Dic3)⋊C3C2312yes—
72(C3⋊Dic3)⋊C3C2312yes—
73D3×#41—91no—
74D3×#41—91no—
75C2×#235—91no—
76C22×C3×D3—31no—
77C22×C3×D3—31no—
78C22×C3×D3—61no—
79C2×D32—32no—
80C2×D32—32no—
81C2×C32×C4—21no—
82C2×C32×C4C2×D311yescharacteristic, complemented
83C2×C32×C4C2×D311yescharacteristic, complemented
84#638C2×D312yescomplemented
85#638C2×D312yescomplemented
86C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
87C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
88C4×C3⋊D3C2×D312yescomplemented
89C4×C3⋊D3C2×D312yescomplemented
90#634C2×D311yescharacteristic, complemented
91#634C2×D311yescharacteristic, complemented
92C3×#67—32no—
93C3×#67—32no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—31no—
96C2×C3×Dic3—61no—
97C3×D12—31no—
98C3×D12—31no—
99C3×C4×D3—32no—
100C3×C4×D3—32no—
101C3×C4×D3—31no—
102C3×C4×D3—31no—
103C3×C4×D3—31no—
104C3×C4×D3—31no—
105C3×C4×D3—31no—
106C3×C4×D3—31no—
107C3×C4×D3—31no—
108C3×C4×D3—31no—
109C3×C4×D3—61no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×Dic6—31no—
114C3×Dic6—31no—
115#627—32no—
116#627—32no—
117#626—32no—
118#626—32no—
119#626—32no—
120#626—32no—
121#625—32no—
122#625—32no—
123#624—32no—
124#624—32no—
125D3×Dic3—32no—
126D3×Dic3—32no—
127D3×Dic3—32no—
128D3×Dic3—32no—
129C2×H(3,𝔽3)—21no—
130C2×H(3,𝔽3)C2411yescharacteristic
131#272—21no—
132#272#4112yescomplemented
133(C3⋊D3)⋊C3—22no—
134(C3⋊D3)⋊C3—22no—
135C3×#41—91no—
136C3×#41—91no—
137C22×C3×C4—91no—
138C2×#67—92no—
139#239—91no—
140#239—91no—
141D3×Q8—91no—
142D3×Q8—91no—
143#237—91no—
144#237—92no—
145#237—92no—
146#237—91no—
147D3×D4—91no—
148D3×D4—91no—
149D3×D4—92no—
150D3×D4—92no—
151#235—31no—
152#235—31no—
153#235—91no—
154#235—91no—
155#235—91no—
156#235—92no—
157#235—92no—
158#235—92no—
159#235—92no—
160#235—181no—
161C2×D12—91no—
162C2×C4×D3—31no—
163C2×C4×D3—31no—
164C2×C4×D3—61no—
165C2×C4×D3—92no—
166C2×C4×D3—92no—
167C2×C4×D3—92no—
168C2×Dic6—91no—
169C22×C32—21no—
170C22×C32C22×D311yescharacteristic
171C22×C32C22×D311yescharacteristic
172C2×C3⋊D3C22×D312yes—
173C2×C3⋊D3C22×D312yes—
174C2×C3×D3—31no—
175C2×C3×D3—31no—
176C2×C3×D3—32no—
177C2×C3×D3—32no—
178C2×C3×D3—31no—
179C2×C3×D3—31no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—64no—
185D32—64no—
186D32—64no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4C22×D311yescharacteristic
190C32×C4C22×D311yescharacteristic
191C32×C4C22×D311yescharacteristic
192C32×C4C22×D311yescharacteristic
193C3⋊Dic3C22×D312yes—
194C3⋊Dic3C22×D312yes—
195C3×Dic3—32no—
196C3×Dic3—32no—
197C3×Dic3—31no—
198C3×Dic3—31no—
199C3×Dic3—31no—
200C3×Dic3—31no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C2×#41—271noSylow
204H(3,𝔽3)C2×#4111yescharacteristic, complemented, Sylow
205C23×C3—91no—
206C22×D3—31no—
207C22×D3—31no—
208C22×D3—61no—
209C22×D3—92no—
210C22×D3—92no—
211C22×D3—92no—
212C3×Q8—91no—
213C3×Q8—91no—
214C3×D4—91no—
215C3×D4—91no—
216C3×D4—92no—
217C3×D4—92no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—121no—
228C2×C3×C4D3211yescharacteristic
229#67—32no—
230#67—32no—
231#67—92no—
232#67—92no—
233#67—92no—
234#67—92no—
235#67—92no—
236#67—92no—
237#67—92no—
238#67—182no—
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—92no—
243C2×Dic3—92no—
244C2×Dic3—92no—
245D12—31no—
246D12—31no—
247D12—91no—
248D12—92no—
249D12—92no—
250D12—92no—
251D12—181no—
252C4×D3—31no—
253C4×D3—31no—
254C4×D3—31no—
255C4×D3—31no—
256C4×D3—31no—
257C4×D3—31no—
258C4×D3—31no—
259C4×D3—31no—
260C4×D3—32no—
261C4×D3—32no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—61no—
265C4×D3—61no—
266C4×D3—92no—
267C4×D3—92no—
268C4×D3—92no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—92no—
272C4×D3—92no—
273C4×D3—182no—
274Dic6—31no—
275Dic6—31no—
276Dic6—91no—
277Dic6—92no—
278Dic6—92no—
279Dic6—92no—
280Dic6—181no—
281C2×C32—21no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32C23×D311yescharacteristic
286C2×C32C23×D311yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—22no—
289C3×D3—32no—
290C3×D3—32no—
291C3×D3—62no—
292C3×D3—62no—
293C3×D3—62no—
294C3×D3—62no—
295C3×D3—61no—
296C3×D3—61no—
297#41—91no—
298#41—91no—
299#41—272no—
300#41—272no—
301#41—271no—
302#41—271no—
303C2×Q8—271no—
304C2×D4—271no—
305C2×D4—272no—
306C22×C4—91no—
307C22×C4—272no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3—92no—
311C22×C3—92no—
312C22×C3—91no—
313C22×C3—91no—
314C22×C3—121no—
315C22×C3—182no—
316C22×C3C2×D3211yescharacteristic
317C2×D3—31no—
318C2×D3—31no—
319C2×D3—31no—
320C2×D3—31no—
321C2×D3—32no—
322C2×D3—32no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—64no—
328C2×D3—92no—
329C2×D3—182no—
330C2×D3—184no—
331C2×D3—184no—
332C2×D3—184no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—92no—
338C3×C4—92no—
339C3×C4—91no—
340C3×C4—91no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4C2×D3211yescharacteristic
344C3×C4C2×D3211yescharacteristic
345Dic3—31no—
346Dic3—31no—
347Dic3—31no—
348Dic3—31no—
349Dic3—32no—
350Dic3—32no—
351Dic3—61no—
352Dic3—61no—
353Dic3—92no—
354Dic3—182no—
355C32—21no—
356C32D3×#4111yescharacteristic, complemented
357C32D3×#4111yescharacteristic, complemented
358C23—91no—
359C23—272no—
360Q8—91no—
361Q8—91no—
362Q8—272no—
363D4—92no—
364D4—92no—
365D4—91no—
366D4—91no—
367D4—272no—
368D4—272no—
369D4—272no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—92no—
376C2×C4—92no—
377C2×C4—272no—
378C2×C4—272no—
379C2×C4—272no—
380C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
381C2×C3—21no—
382C2×C3—31no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—92no—
386C2×C3—121no—
387C2×C3—122no—
388C2×C3—121no—
389C2×C3—182no—
390C2×C3—182no—
391C2×C3—181no—
392C2×C3C22×D3211yescharacteristic
393D3—32no—
394D3—32no—
395D3—62no—
396D3—61no—
397D3—61no—
398D3—62no—
399D3—62no—
400D3—62no—
401D3—182no—
402D3—362no—
403C22—91no—
404C22—91no—
405C22—92no—
406C22—92no—
407C22—182no—
408C22—544no—
409C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
410C4—91no—
411C4—91no—
412C4—92no—
413C4—92no—
414C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C3—31no—
417C3—61no—
418C3—121no—
419C3#6430611yescharacteristic
420C2—21no—
421C2—92no—
422C2—181no—
423C2—182no—
424C2—182no—
425C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
Frattini, socleC2×C3C22×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×#272C2semidirect
2. #3828⋊C3C2semidirect
3. #3828⋊C3C2semidirect
4. #88677C2semidirect
5. #88672C2semidirect
6. #88655C2semidirect
7. C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
8. #88651C2semidirect
9. #88650C2semidirect
10. #88649C2semidirect
11. #88649C2semidirect
16. C2×C4×H(3,𝔽3)C22semidirect
17. C2×#1084C22semidirect
20. C4×#272C22semidirect
21. C4×#272C22semidirect
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. #634⋊C3C22semidirect
29. #634⋊C3C22semidirect
30. #6200C22semidirect
31. #6200C22semidirect
33. #6198C22semidirect
34. ((C3⋊D3)⋊C3)⋊C4C22semidirect
35. ((C3⋊D3)⋊C3)⋊C4C22semidirect
36. #6196C22semidirect
37. #3828D3semidirect
38. #3828D3semidirect
82. C2×C32×C4C2×D3semidirect
83. C2×C32×C4C2×D3semidirect
84. #638C2×D3semidirect
85. #638C2×D3semidirect
86. C4⋊(C3⋊D3)C2×D3semidirect
87. C4⋊(C3⋊D3)C2×D3semidirect
88. C4×C3⋊D3C2×D3semidirect
89. C4×C3⋊D3C2×D3semidirect
90. #634C2×D3semidirect
91. #634C2×D3semidirect
132. #272#41semidirect
204. H(3,𝔽3)C2×#41semidirect
356. C32D3×#41semidirect
357. C32D3×#41semidirect
380. C2×C4((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
#3828⋊C32
#886772
#886722
#886552
C4×((C3⋊D3)⋊C3)⋊C22
#886512
#886502
#886492
D3×#2353