Finite Group Databasebeta

← #11722924

Subgroups — #11722924

Order 864.

506 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 102 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×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#625C211yesmaximal, characteristic
3C2×C3×D3×Dic3C211yesmaximal, characteristic, complemented
4#88813C211yesmaximal, characteristic, complemented
5#88807C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#67⋊Dic3—31nomaximal
9Dic3×#67—31nomaximal
10#63879—31nomaximal
11C22×C32×D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18D3×C3⋊Dic3—22no—
19C3×#625C414yescomplemented
20C3×D3×Dic3—22no—
21C22×C3⋊Dic3—31no—
22C2×C3×#67—31no—
23C2×C3×#67—31no—
24C22×C3×Dic3—31no—
25C2×C3×C4×D3—31no—
26C2×#625D311yescharacteristic
27C2×D3×Dic3—31no—
28C2×D3×Dic3—31no—
29C2×D3×Dic3—61no—
30C2×D3×Dic3D311yescharacteristic, complemented
31#3757—31no—
32#3750—31no—
33C3×#217—31no—
34C3×(C2×D3)⋊C4—31no—
35Dic3⋊Dic3—31no—
36Dic3⋊Dic3—31no—
37Dic3⋊Dic3—61no—
38(C2×D3)⋊Dic3—31no—
39(C2×D3)⋊Dic3—31no—
40(C2×D3)⋊Dic3—61no—
41(C2×D3)⋊Dic3D311yescharacteristic, complemented
42Dic32—31no—
43Dic32—31no—
44Dic32—31no—
45C22×C33C2311yescharacteristic
46C2×C32×D3—21no—
47C2×C32×D3—21no—
48C2×C32×D3—21no—
49C2×C32×D3—21no—
50C2×C32×D3C2×C412yescomplemented
51C2×C32×D3C2×C412yescomplemented
52C3⋊(C3⋊Dic3)—21no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3C2×C412yescomplemented
55C3×C3⋊Dic3D412yescomplemented
56C32×Dic3—21no—
57D4×Dic3—91no—
58C4×#67—91no—
59#67⋊C4—91no—
60C23×C32—31no—
61C23×C32—31no—
62C22×C3×D3—21no—
63C22×C3×D3—21no—
64C22×C3×D3C2×D311yescharacteristic
65C22×C3×D3C2×D311yescharacteristic
66C22×C3×D3C2×D311yescharacteristic
67C22×C3×D3C2×D311yescharacteristic
68C2×C32×C4—31no—
69C2×C3⋊Dic3—31no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—61no—
75C2×C3⋊Dic3—62no—
76C2×C3⋊Dic3—121no—
77C2×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊Dic3C2×D311yescharacteristic
79C2×C3⋊Dic3C2×D311yescharacteristic
80C3×#67—34no—
81C3×#67—34no—
82C2×C3×Dic3—21no—
83C2×C3×Dic3—31no—
84C2×C3×Dic3—31no—
85C2×C3×Dic3—31no—
86C2×C3×Dic3—31no—
87C2×C3×Dic3—31no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3C2×D311yescharacteristic
94C2×C3×Dic3C2×D311yescharacteristic
95C3×C4×D3—62no—
96#625Dic314yescomplemented
97D3×Dic3—22no—
98D3×Dic3—62no—
99D3×Dic3—62no—
100D3×Dic3—64no—
101C2×C33C22×C411yescharacteristic
102C2×C33C2×D411yescharacteristic
103C2×C33#4111yescharacteristic
104C32×D3—22no—
105C32×D3—22no—
106C2×C3×D4—91no—
107C22×C3×C4—91no—
108C2×#67—31no—
109C2×#67—31no—
110C22×Dic3—31no—
111C22×Dic3—91no—
112C22×Dic3—91no—
113C22×Dic3—181no—
114C2×C4×D3—31no—
115C2×C4×D3—91no—
116C3×C22⋊C4—91no—
117#217—31no—
118#217—91no—
119#217—91no—
120#217—181no—
121(C2×D3)⋊C4—31no—
122(C2×D3)⋊C4—91no—
123C4⋊Dic3—91no—
124Dic3⋊C4—91no—
125Dic3⋊C4—91no—
126Dic3⋊C4—181no—
127C4×Dic3—91no—
128C4×Dic3—91no—
129C4×Dic3—91no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—32no—
134C22×C32—32no—
135C22×C32—41no—
136C22×C32—61no—
137C22×C32—61no—
138C22×C32—61no—
139C22×C32—61no—
140C22×C32C22×D311yescharacteristic
141C22×C32C22×D311yescharacteristic
142C22×C32C22×D311yescharacteristic
143C2×C3×D3—22no—
144C2×C3×D3—22no—
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—21no—
153C2×C3×D3—22no—
154C2×C3×D3—22no—
155C2×C3×D3—41no—
156C2×C3×D3—41no—
157C2×C3×D3C4×D312yescomplemented
158C2×C3×D3C4×D312yescomplemented
159C2×C3×D3C2×Dic312yescomplemented
160C2×C3×D3C2×Dic312yescomplemented
161C32×C4—61no—
162C3⋊Dic3—21no—
163C3⋊Dic3—61no—
164C3⋊Dic3—62no—
165C3⋊Dic3—61no—
166C3⋊Dic3—61no—
167C3⋊Dic3—121no—
168C3⋊Dic3—121no—
169C3⋊Dic3—122no—
170C3⋊Dic3C2×Dic312yescomplemented
171C3⋊Dic3#6712yescomplemented
172C3×Dic3—21no—
173C3×Dic3—21no—
174C3×Dic3—32no—
175C3×Dic3—32no—
176C3×Dic3—32no—
177C3×Dic3—32no—
178C3×Dic3—41no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—121no—
184C4×D4—271noSylow
185C33C4×D411yescharacteristic, complemented, Sylow
186C23×C3—31no—
187C23×C3—31no—
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—61no—
191C23×C3—61no—
192C22×D3D3211yescharacteristic
193C22×D3D3211yescharacteristic
194C3×D4—94no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—182no—
202#67—34no—
203#67—34no—
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209C2×Dic3—31no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—91no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—184no—
223C2×Dic3—361no—
224C2×Dic3D3211yescharacteristic
225C4×D3—62no—
226C4×D3—182no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—62no—
240C2×C32—62no—
241C2×C32C2×C4×D311yescharacteristic
242C2×C32C2×C4×D311yescharacteristic
243C2×C32#23511yescharacteristic
244C2×C32D3×D411yescharacteristic
245C2×C32D3×D411yescharacteristic
246C2×C32#23711yescharacteristic
247C2×C32#23711yescharacteristic
248C2×C32C22×Dic311yescharacteristic
249C2×C32C2×#6711yescharacteristic
250C3×D3—24no—
251C3×D3—22no—
252C3×D3—22no—
253C3×D3—22no—
254C3×D3—22no—
255C3×D3—42no—
256C2×D4—91no—
257C22×C4—91no—
258C22×C4—271no—
259C4⋊C4—271no—
260C22⋊C4—91no—
261C22⋊C4—271no—
262C42—271no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—41no—
271C22×C3—62no—
272C22×C3—62no—
273C22×C3—62no—
274C22×C3—62no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—61no—
282C22×C3—61no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3C2×D3211yescharacteristic
286C22×C3C2×D3211yescharacteristic
287C22×C3C2×D3211yescharacteristic
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—21no—
292C2×D3D3×Dic312yescomplemented
293C2×D3D3×Dic312yescomplemented
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—92no—
297C3×C4—92no—
298C3×C4—121no—
299C3×C4—181no—
300Dic3—21no—
301Dic3—32no—
302Dic3—32no—
303Dic3—32no—
304Dic3—32no—
305Dic3—62no—
306Dic3—62no—
307Dic3—61no—
308Dic3—61no—
309Dic3—121no—
310Dic3—182no—
311Dic3—181no—
312Dic3—181no—
313Dic3—181no—
314Dic3—361no—
315Dic3—361no—
316Dic3—362no—
317C32—21no—
318C32—21no—
319C32—21no—
320C32—41no—
321C32#67⋊C411yescharacteristic, complemented
322C32C4×#6711yescharacteristic, complemented
323C32D4×Dic311yescharacteristic, complemented
324C23—31no—
325C23—31no—
326D4—94no—
327C2×C4—31no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—182no—
332C2×C4—271no—
333C2×C4—542no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—64no—
347C2×C3—62no—
348C2×C3—62no—
349C2×C3—62no—
350C2×C3—62no—
351C2×C3—122no—
352C2×C3#379511yescharacteristic
353C2×C3C4×D3211yescharacteristic
354C2×C3C2×D3×Dic311yescharacteristic
355C2×C3C2×D3×Dic311yescharacteristic
356C2×C3#380411yescharacteristic
357C2×C3#380511yescharacteristic
358C2×C3D3×#6711yescharacteristic
359C2×C3D3×#6711yescharacteristic
360C2×C3#381111yescharacteristic
361D3—22no—
362D3—22no—
363C22—32no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22D3311yescharacteristic
370C4—61no—
371C4—92no—
372C4—92no—
373C4—181no—
374C4—541no—
375C3—21no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3#6387911yescharacteristic, complemented
380C3Dic3×#6711yescharacteristic, complemented
381C3#67⋊Dic311yescharacteristic, complemented
382C2—62no—
383C2—62no—
384C2D32×Dic311yescharacteristic
385C2#8895211yescharacteristic
386C2#8896111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutatorC2×C33C22×C4
Fitting, socle, cosocleC22×C33C23
prime radicalC2×C3×#625C2
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊Dic3C2semidirect
3. C2×C3×D3×Dic3C2semidirect
4. #88813C2semidirect
5. #88807C2semidirect
6. (C2×D3)⋊(C3⋊Dic3)C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
19. C3×#625C4semidirect
30. C2×D3×Dic3D3semidirect
41. (C2×D3)⋊Dic3D3semidirect
50. C2×C32×D3C2×C4semidirect
51. C2×C32×D3C2×C4semidirect
54. C3×C3⋊Dic3C2×C4semidirect
55. C3×C3⋊Dic3D4semidirect
96. #625Dic3semidirect
157. C2×C3×D3C4×D3semidirect
158. C2×C3×D3C4×D3semidirect
159. C2×C3×D3C2×Dic3semidirect
160. C2×C3×D3C2×Dic3semidirect
170. C3⋊Dic3C2×Dic3semidirect
171. C3⋊Dic3#67semidirect
185. C33C4×D4semidirect
292. C2×D3D3×Dic3semidirect
293. C2×D3D3×Dic3semidirect
321. C32#67⋊C4semidirect
322. C32C4×#67semidirect
323. C32D4×Dic3semidirect
379. C3#63879semidirect
380. C3Dic3×#67semidirect
381. C3#67⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊Dic32
C2×C3×#6252
C2×C3×D3×Dic32
#888132
#888072
(C2×D3)⋊(C3⋊Dic3)2
C3×(C2×D3)⋊Dic32
#67⋊Dic33
Dic3×#673
#638793