Finite Group Databasebeta

← #11722966

Subgroups — #11722966

Order 864.

550 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 78 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×#6294C211yesmaximal, characteristic, complemented
2C2×#6292C211yesmaximal, characteristic, complemented
3C2×#6291C211yesmaximal, characteristic, complemented
4C2×C3×#626C211yesmaximal, characteristic, complemented
5#88804C211yesmaximal, characteristic, complemented
6C3×#3724C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63971—31nomaximal
9#63910—31nomaximal
10#63886—31nomaximal
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C22×C32×D3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6294—22no—
19#6292—22no—
20#6291—22no—
21C3×#626—22no—
22C2×#638—31no—
23C2×C4⋊(C3⋊D3)—31no—
24C2×C3×#67—31no—
25C2×C3×D12—31no—
26C2×#626—31no—
27C2×#626—31no—
28C2×#626—31no—
29C2×#626—31no—
30C2×#626—61no—
31C2×#626—61no—
32C2×#626D311yescharacteristic, complemented
33C2×#624—31no—
34C2×D3×Dic3—31no—
35C2×D3×Dic3—31no—
36#3752—31no—
37C3×#217—31no—
38C3×(C2×D3)⋊C4—31no—
39C3×Dic3⋊C4—31no—
40C3×Dic3⋊C4—31no—
41#3724D311yescharacteristic, complemented
42#3722—31no—
43#3722—31no—
44#3722—61no—
45(C2×D3)⋊Dic3D311yescharacteristic, complemented
46C22×C33C2311yescharacteristic
47C2×C3⋊(C3⋊D3)—21no—
48C2×C3⋊(C3⋊D3)—21no—
49C2×C3⋊(C3⋊D3)—21no—
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3D412yescomplemented
53C2×C32×D3—21no—
54C2×C32×D3—21no—
55C2×C32×D3—21no—
56C3×C3⋊Dic3—21no—
57C3×C3⋊Dic3D412yescomplemented
58C32×Dic3—21no—
59C32×Dic3—21no—
60#939—91no—
61#895—91no—
62#884—91no—
63C23×C32—31no—
64C22×C3⋊D3—31no—
65C22×C3⋊D3—31no—
66C22×C3⋊D3—31no—
67C22×C3⋊D3—61no—
68C22×C3⋊D3—61no—
69C22×C3⋊D3—61no—
70C22×C3⋊D3—121no—
71C22×C3⋊D3C2×D311yescharacteristic
72C22×C3×D3—21no—
73C22×C3×D3—31no—
74C22×C3×D3—31no—
75C22×C3×D3—61no—
76C22×C3×D3C2×D311yescharacteristic
77C22×C3×D3C2×D311yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80#638—62no—
81C2×C3⋊Dic3C2×D311yescharacteristic
82C2×C3⋊Dic3C2×D311yescharacteristic
83C4⋊(C3⋊D3)—62no—
84C3×#67—62no—
85C2×C3×Dic3—21no—
86C2×C3×Dic3—21no—
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×Dic3C2×D311yescharacteristic
94C2×C3×Dic3C2×D311yescharacteristic
95C2×C3×Dic3C2×D311yescharacteristic
96C2×C3×Dic3C2×D311yescharacteristic
97C3×D12—62no—
98#626—22no—
99#626—62no—
100#626—62no—
101#626—64no—
102#626—62no—
103#626—62no—
104#626—64no—
105#624—62no—
106D3×Dic3—62no—
107D3×Dic3—62no—
108C2×C33C2×D411yescharacteristic
109C2×C33C2×D411yescharacteristic
110C2×C33#4111yescharacteristic
111C3⋊(C3⋊D3)—41no—
112C3×C3⋊D3—22no—
113C32×D3—41no—
114C2×C3×D4—91no—
115C2×#67—31no—
116C2×#67—91no—
117C2×#67—91no—
118C2×#67—91no—
119C2×#67—181no—
120C22×Dic3—91no—
121C2×D12—31no—
122C2×D12—91no—
123C2×D12—91no—
124C2×D12—91no—
125C2×D12—181no—
126C2×C4×D3—91no—
127C2×C4×D3—91no—
128C3×C4⋊C4—91no—
129C3×C22⋊C4—91no—
130#217—31no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—91no—
133(C2×D3)⋊C4—91no—
134(C2×D3)⋊C4—91no—
135(C2×D3)⋊C4—181no—
136Dic3⋊C4—31no—
137Dic3⋊C4—31no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—61no—
143C22×C32—61no—
144C22×C32—61no—
145C22×C32C22×D311yescharacteristic
146C22×C32C22×D311yescharacteristic
147C22×C32C22×D311yescharacteristic
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—61no—
151C2×C3⋊D3—61no—
152C2×C3⋊D3—61no—
153C2×C3⋊D3—62no—
154C2×C3⋊D3—62no—
155C2×C3⋊D3—62no—
156C2×C3⋊D3—61no—
157C2×C3⋊D3—61no—
158C2×C3⋊D3—61no—
159C2×C3⋊D3—61no—
160C2×C3⋊D3—61no—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—121no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—121no—
166C2×C3⋊D3—121no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—122no—
169C2×C3⋊D3—122no—
170C2×C3⋊D3—122no—
171C2×C3⋊D3#6712yescomplemented
172C2×C3×D3—22no—
173C2×C3×D3—21no—
174C2×C3×D3—21no—
175C2×C3×D3—21no—
176C2×C3×D3—21no—
177C2×C3×D3—21no—
178C2×C3×D3—21no—
179C2×C3×D3—32no—
180C2×C3×D3—32no—
181C2×C3×D3—41no—
182C2×C3×D3—41no—
183C2×C3×D3—62no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—61no—
191C32×C4—61no—
192C3⋊Dic3—21no—
193C3⋊Dic3D1212yescomplemented
194C3×Dic3—22no—
195C3×Dic3—21no—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—32no—
200C3×Dic3—32no—
201C3×Dic3—41no—
202C3×Dic3—62no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—121no—
206#121—271noSylow
207C33#12111yescharacteristic, complemented, Sylow
208C23×C3—31no—
209C23×C3—31no—
210C23×C3—61no—
211C23×C3—91no—
212C22×D3—31no—
213C22×D3—31no—
214C22×D3—61no—
215C22×D3—91no—
216C22×D3—91no—
217C22×D3—91no—
218C22×D3—181no—
219C22×D3—181no—
220C22×D3—181no—
221C22×D3—361no—
222C22×D3D3211yescharacteristic
223C3×D4—182no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232#67—62no—
233#67—182no—
234#67—182no—
235#67—182no—
236#67—184no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—182no—
244C2×Dic3D3211yescharacteristic
245C2×Dic3D3211yescharacteristic
246D12—62no—
247D12—182no—
248D12—182no—
249D12—184no—
250D12—182no—
251C4×D3—182no—
252C4×D3—182no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—121no—
266C2×C32C2×D1211yescharacteristic
267C2×C32#23511yescharacteristic
268C2×C32D3×D411yescharacteristic
269C2×C32D3×D411yescharacteristic
270C2×C32D3×D411yescharacteristic
271C2×C32D3×D411yescharacteristic
272C2×C32#23711yescharacteristic
273C2×C32#23911yescharacteristic
274C2×C32C2×#6711yescharacteristic
275C3⋊D3—22no—
276C3⋊D3—122no—
277C3⋊D3—122no—
278C3⋊D3—122no—
279C3⋊D3—124no—
280C3⋊D3—121no—
281C3⋊D3—121no—
282C3⋊D3—121no—
283C3×D3—42no—
284C3×D3—41no—
285C3×D3—41no—
286C3×D3—62no—
287C3×D3—62no—
288C3×D3—122no—
289C2×D4—91no—
290C2×D4—271no—
291C2×D4—271no—
292C22×C4—271no—
293C4⋊C4—91no—
294C22⋊C4—91no—
295C22⋊C4—271no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—92no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—181no—
311C22×C3—181no—
312C22×C3C2×D3211yescharacteristic
313C22×C3C2×D3211yescharacteristic
314C22×C3C2×D3211yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—32no—
319C2×D3—32no—
320C2×D3—62no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—182no—
328C2×D3—182no—
329C2×D3—182no—
330C2×D3—181no—
331C2×D3—181no—
332C2×D3—181no—
333C2×D3—181no—
334C2×D3—181no—
335C2×D3—181no—
336C2×D3—181no—
337C2×D3—181no—
338C2×D3—181no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C2×D3—362no—
346C2×D3—362no—
347C2×D3—362no—
348C3×C4—62no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—92no—
354C3×C4—121no—
355C3×C4—181no—
356Dic3—21no—
357Dic3—21no—
358Dic3—32no—
359Dic3—32no—
360Dic3—62no—
361Dic3—61no—
362Dic3—61no—
363Dic3—121no—
364C32—21no—
365C32—21no—
366C32—21no—
367C32—41no—
368C32#88411yescharacteristic, complemented
369C32#89511yescharacteristic, complemented
370C32#93911yescharacteristic, complemented
371C23—31no—
372C23—91no—
373C23—271no—
374D4—182no—
375D4—542no—
376D4—542no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—91no—
380C2×C4—91no—
381C2×C4—542no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—122no—
395C2×C3—121no—
396C2×C3—121no—
397C2×C3—182no—
398C2×C3#379611yescharacteristic
399C2×C3#379911yescharacteristic
400C2×C3D3×D1211yescharacteristic
401C2×C3D3×D1211yescharacteristic
402C2×C3#380411yescharacteristic
403C2×C3D3×#6711yescharacteristic
404C2×C3D3×#6711yescharacteristic
405C2×C3#381111yescharacteristic
406C2×C3#381111yescharacteristic
407D3—41no—
408D3—62no—
409D3—62no—
410D3—122no—
411D3—362no—
412D3—362no—
413D3—362no—
414D3—364no—
415D3—361no—
416D3—361no—
417D3—361no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—92no—
422C22—181no—
423C22—181no—
424C22—541no—
425C22—541no—
426C22—541no—
427C22D3311yescharacteristic
428C4—61no—
429C4—61no—
430C4—92no—
431C4—181no—
432C3—21no—
433C3—21no—
434C3—21no—
435C3—41no—
436C3#6388611yescharacteristic, complemented
437C3#6391011yescharacteristic, complemented
438C3#6397111yescharacteristic, complemented
439C2—121no—
440C2—182no—
441C2—1081no—
442C2#8895211yescharacteristic
443C2#8895511yescharacteristic
444C2#8896211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6294C2semidirect
2. C2×#6292C2semidirect
3. C2×#6291C2semidirect
4. C2×C3×#626C2semidirect
5. #88804C2semidirect
6. C3×#3724C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
32. C2×#626D3semidirect
41. #3724D3semidirect
45. (C2×D3)⋊Dic3D3semidirect
52. C2×C3×C3⋊D3D4semidirect
57. C3×C3⋊Dic3D4semidirect
171. C2×C3⋊D3#67semidirect
193. C3⋊Dic3D12semidirect
207. C33#121semidirect
368. C32#884semidirect
369. C32#895semidirect
370. C32#939semidirect
436. C3#63886semidirect
437. C3#63910semidirect
438. C3#63971semidirect

Maximal subgroups

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

subgroupindex
C2×#62942
C2×#62922
C2×#62912
C2×C3×#6262
#888042
C3×#37242
C3×(C2×D3)⋊Dic32
#639713
#639103
#638863