Finite Group Databasebeta

← #11723726

Subgroups — #11723726

Order 864.

550 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 107 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×C3×#638C211yesmaximal, characteristic
2C22×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#6294C211yesmaximal, characteristic, complemented
4C3×#3757C211yesmaximal, characteristic, complemented
5#88815C211yesmaximal, characteristic, complemented
6#88814C211yesmaximal, characteristic, complemented
7#88813C211yesmaximal, characteristic, complemented
8#67⋊Dic3—31nomaximal
9Dic3×#67—31nomaximal
10#63952—31nomaximal
11C23×C33C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C3×#638C414yescomplemented
14C2×C3×C3⋊Dic3—22no—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊Dic3C2211yescharacteristic
20#6294—22no—
21C2×#638D311yescharacteristic
22C22×C3⋊Dic3D311yescharacteristic, complemented
23C2×C3×#67—31no—
24C2×C3×#67—31no—
25C2×C3×#67—61no—
26C22×C3×Dic3—31no—
27C22×C3×Dic3—31no—
28C22×C3×Dic3—61no—
29C2×#624—31no—
30C2×D3×Dic3—31no—
31C2×D3×Dic3—31no—
32#3757D311yescharacteristic, complemented
33C3×#217—31no—
34C3×#217—31no—
35C3×#217—61no—
36#3724—31no—
37Dic3⋊Dic3—31no—
38Dic3⋊Dic3—31no—
39#3722—31no—
40(C2×D3)⋊Dic3—31no—
41(C2×D3)⋊Dic3—31no—
42Dic32—31no—
43Dic32—31no—
44Dic32—31no—
45C22×C33—21no—
46C22×C33—21no—
47C22×C33C2×C412yescomplemented
48C22×C33C2311yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3C2×C412yescomplemented
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3C2×C412yescomplemented
56C3×C3⋊Dic3D412yescomplemented
57D4×Dic3—91no—
58C4×#67—91no—
59#67⋊C4—91no—
60C23×C32—21no—
61C23×C32—21no—
62C23×C32—21no—
63C23×C32—41no—
64C23×C32C2×D311yescharacteristic
65C23×C32C2×D311yescharacteristic
66C23×C32C2×D311yescharacteristic
67C22×C3⋊D3C2×D311yescharacteristic
68C22×C3×D3—31no—
69C22×C3×D3—31no—
70C22×C3×D3—61no—
71#638Dic314yescomplemented
72C2×C3⋊Dic3—22no—
73C2×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊Dic3C2×D311yescharacteristic
76C2×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊Dic3C2×D311yescharacteristic
78C3×#67—34no—
79C3×#67—34no—
80C3×#67—64no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—31no—
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—31no—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—64no—
99#624—62no—
100D3×Dic3—62no—
101D3×Dic3—62no—
102C2×C33—22no—
103C2×C33C22×C411yescharacteristic
104C2×C33C2×D411yescharacteristic
105C2×C33#4111yescharacteristic
106C3×C3⋊D3—22no—
107C2×C3×D4—91no—
108C22×C3×C4—91no—
109C2×#67—31no—
110C2×#67—31no—
111C2×#67—61no—
112C22×Dic3—31no—
113C22×Dic3—31no—
114C22×Dic3—61no—
115C22×Dic3—91no—
116C2×C4×D3—91no—
117C2×C4×D3—91no—
118C3×C22⋊C4—91no—
119#217—31no—
120#217—31no—
121#217—61no—
122#217—91no—
123(C2×D3)⋊C4—91no—
124(C2×D3)⋊C4—91no—
125C4⋊Dic3—91no—
126Dic3⋊C4—91no—
127Dic3⋊C4—91no—
128C4×Dic3—91no—
129C4×Dic3—91no—
130C4×Dic3—91no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—42no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32C4×D312yescomplemented
154C22×C32C4×D312yescomplemented
155C22×C32C2×Dic312yescomplemented
156C22×C32C22×D311yescharacteristic
157C22×C32C22×D311yescharacteristic
158C22×C32C22×D311yescharacteristic
159C2×C3⋊D3—21no—
160C2×C3⋊D3—21no—
161C2×C3⋊D3C2×Dic312yescomplemented
162C2×C3×D3—32no—
163C2×C3×D3—32no—
164C2×C3×D3—62no—
165C2×C3×D3—61no—
166C2×C3×D3—61no—
167C2×C3×D3—61no—
168C2×C3×D3—61no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C3⋊Dic3—21no—
172C3⋊Dic3—21no—
173C3⋊Dic3—21no—
174C3⋊Dic3C2×Dic312yescomplemented
175C3⋊Dic3#6712yescomplemented
176C3×Dic3—32no—
177C3×Dic3—32no—
178C3×Dic3—32no—
179C3×Dic3—32no—
180C3×Dic3—62no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C3×Dic3—121no—
190C3×Dic3—121no—
191C4×D4—271noSylow
192C33C4×D411yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—41no—
197C23×C3—91no—
198C23×C3D3211yescharacteristic
199C23×C3D3211yescharacteristic
200C23×C3D3211yescharacteristic
201C22×D3—31no—
202C22×D3—31no—
203C22×D3—61no—
204C3×D4—94no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—182no—
211#67—34no—
212#67—34no—
213#67—64no—
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—64no—
232C2×Dic3—182no—
233C4×D3—182no—
234C4×D3—182no—
235C2×C32—22no—
236C2×C32—24no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32—42no—
253C2×C32—44no—
254C2×C32C2×C4×D311yescharacteristic
255C2×C32C2×C4×D311yescharacteristic
256C2×C32#23511yescharacteristic
257C2×C32D3×D411yescharacteristic
258C2×C32D3×D411yescharacteristic
259C2×C32#23711yescharacteristic
260C2×C32#23711yescharacteristic
261C2×C32C22×Dic311yescharacteristic
262C2×C32C2×#6711yescharacteristic
263C3⋊D3—22no—
264C3×D3—62no—
265C3×D3—62no—
266C3×D3—122no—
267C2×D4—91no—
268C22×C4—91no—
269C22×C4—271no—
270C4⋊C4—271no—
271C22⋊C4—91no—
272C22⋊C4—271no—
273C42—271no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—92no—
297C22×C3—181no—
298C22×C3—181no—
299C22×C3D3×Dic312yescomplemented
300C22×C3D3×Dic312yescomplemented
301C22×C3#62412yescomplemented
302C22×C3C2×D3211yescharacteristic
303C22×C3C2×D3211yescharacteristic
304C22×C3C2×D3211yescharacteristic
305C2×D3—32no—
306C2×D3—32no—
307C2×D3—62no—
308C2×D3—61no—
309C2×D3—61no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—121no—
313C2×D3—121no—
314C3×C4—92no—
315C3×C4—92no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—181no—
319Dic3—32no—
320Dic3—32no—
321Dic3—32no—
322Dic3—32no—
323Dic3—62no—
324Dic3—62no—
325Dic3—62no—
326Dic3—61no—
327Dic3—61no—
328Dic3—61no—
329Dic3—61no—
330Dic3—61no—
331Dic3—61no—
332Dic3—121no—
333Dic3—121no—
334C32—21no—
335C32—21no—
336C32—21no—
337C32—41no—
338C32#67⋊C411yescharacteristic, complemented
339C32C4×#6711yescharacteristic, complemented
340C32D4×Dic311yescharacteristic, complemented
341C23—91no—
342C23#111311yescharacteristic
343D4—94no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—182no—
350C2×C4—542no—
351C2×C3—24no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—42no—
365C2×C3—42no—
366C2×C3—44no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—182no—
371C2×C3C2×D3×Dic311yescharacteristic
372C2×C3C2×D3×Dic311yescharacteristic
373C2×C3#380411yescharacteristic
374C2×C3#380411yescharacteristic
375C2×C3#380511yescharacteristic
376C2×C3C2×#62411yescharacteristic
377C2×C3D3×#6711yescharacteristic
378C2×C3D3×#6711yescharacteristic
379C2×C3#381111yescharacteristic
380D3—62no—
381D3—62no—
382D3—122no—
383C22—21no—
384C22—21no—
385C22—92no—
386C22—181no—
387C22—181no—
388C22#629412yescomplemented
389C22C2×#111311yescharacteristic
390C4—92no—
391C4—92no—
392C4—181no—
393C4—181no—
394C4—181no—
395C3—21no—
396C3—21no—
397C3—21no—
398C3—41no—
399C3#6395211yescharacteristic, complemented
400C3Dic3×#6711yescharacteristic, complemented
401C3#67⋊Dic311yescharacteristic, complemented
402C2—22no—
403C2—182no—
404C2C2×#629411yescharacteristic
405C2#8904611yescharacteristic
406C2#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C2×#1113
commutatorC2×C33C22×C4
FittingC23×C33C22
socle, cosocleC22×C33C23
prime radicalC2×C3×#638C2
hypercenterC23#1113
hypocenterC33C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊Dic3C2semidirect
3. C2×#6294C2semidirect
4. C3×#3757C2semidirect
5. #88815C2semidirect
6. #88814C2semidirect
7. #88813C2semidirect
13. C3×#638C4semidirect
22. C22×C3⋊Dic3D3semidirect
32. #3757D3semidirect
47. C22×C33C2×C4semidirect
51. C2×C3×C3⋊D3C2×C4semidirect
55. C3×C3⋊Dic3C2×C4semidirect
56. C3×C3⋊Dic3D4semidirect
71. #638Dic3semidirect
153. C22×C32C4×D3semidirect
154. C22×C32C4×D3semidirect
155. C22×C32C2×Dic3semidirect
161. C2×C3⋊D3C2×Dic3semidirect
174. C3⋊Dic3C2×Dic3semidirect
175. C3⋊Dic3#67semidirect
192. C33C4×D4semidirect
299. C22×C3D3×Dic3semidirect
300. C22×C3D3×Dic3semidirect
301. C22×C3#624semidirect
338. C32#67⋊C4semidirect
339. C32C4×#67semidirect
340. C32D4×Dic3semidirect
388. C22#6294semidirect
399. C3#63952semidirect
400. C3Dic3×#67semidirect
401. C3#67⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C22×C3×C3⋊Dic32
C2×#62942
C3×#37572
#888152
#888142
#888132
#67⋊Dic33
Dic3×#673
#639523