Finite Group Databasebeta

← #11723009

Subgroups — #11723009

Order 864.

442 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 72 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×#6293C211yesmaximal, characteristic, complemented
2C2×#6290C211yesmaximal, characteristic
3#88814C211yesmaximal, characteristic, complemented
4Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#3722C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63953—31nomaximal
9#63915—31nomaximal
10#63913—31nomaximal
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6293—22no—
19#6290—22no—
20C2×#638—31no—
21C2×#634—31no—
22C2×#627—31no—
23C2×#627—31no—
24C2×#627—61no—
25C2×#625—31no—
26C2×#625—31no—
27C2×#625—61no—
28C4×C3⋊Dic3—31no—
29C3×#217—31no—
30C3×#217—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—31no—
33C3×(C2×D3)⋊C4—31no—
34C3×(C2×D3)⋊C4—31no—
35#3722—31no—
36#3722D311yescharacteristic, complemented
37(C2×D3)⋊Dic3—31no—
38(C2×D3)⋊Dic3—31no—
39(C2×D3)⋊Dic3D311yescharacteristic, complemented
40(C2×D3)⋊Dic3D311yescharacteristic, complemented
41Dic32—31no—
42Dic32—31no—
43Dic32—61no—
44C22×C33C2311yescharacteristic
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—21no—
48C2×C32×D3—21no—
49C2×C32×D3—21no—
50C2×C32×D3—21no—
51C3⋊(C3⋊Dic3)D412yes—
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—21no—
54C32×Dic3—21no—
55C32×Dic3—21no—
56#885—91no—
57#885—91no—
58#885—91no—
59C23×C32—31no—
60C22×C3⋊D3C2×D311yescharacteristic
61C22×C3×D3—21no—
62C22×C3×D3—31no—
63C22×C3×D3—31no—
64C22×C3×D3—61no—
65C22×C3×D3C2×D311yescharacteristic
66C22×C3×D3C2×D311yescharacteristic
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69#638—62no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—31no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—61no—
75C2×C3⋊Dic3—61no—
76C2×C3⋊Dic3—121no—
77C2×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊Dic3C2×D311yescharacteristic
79#634—62no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—21no—
82C2×C3×Dic3—31no—
83C2×C3×Dic3—31no—
84C2×C3×Dic3—31no—
85C2×C3×Dic3—31no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3C2×D311yescharacteristic
89C2×C3×Dic3C2×D311yescharacteristic
90C2×C3×Dic3C2×D311yescharacteristic
91C2×C3×Dic3C2×D311yescharacteristic
92#627—62no—
93#627—62no—
94#627—64no—
95#625—62no—
96#625—62no—
97#625—64no—
98C2×C33C2×D411yescharacteristic
99C2×C33#4111yescharacteristic
100C2×C33#4111yescharacteristic
101C3×C3⋊D3—41no—
102C32×D3—41no—
103C2×#67—91no—
104C2×#67—91no—
105C2×#67—91no—
106C2×#67—181no—
107C2×Dic6—91no—
108C2×Dic6—91no—
109C2×Dic6—91no—
110C2×Dic6—181no—
111C3×C22⋊C4—91no—
112C3×C22⋊C4—91no—
113C3×C22⋊C4—91no—
114#217—31no—
115#217—31no—
116#217—91no—
117(C2×D3)⋊C4—31no—
118(C2×D3)⋊C4—31no—
119(C2×D3)⋊C4—31no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—91no—
122(C2×D3)⋊C4—91no—
123C4×Dic3—91no—
124C4×Dic3—91no—
125C4×Dic3—91no—
126C4×Dic3—181no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—41no—
131C22×C32—61no—
132C22×C32—61no—
133C22×C32—61no—
134C22×C32C22×D311yescharacteristic
135C22×C32C22×D311yescharacteristic
136C22×C32C22×D311yescharacteristic
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3×D3—22no—
141C2×C3×D3—21no—
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3—41no—
148C2×C3×D3—41no—
149C2×C3×D3—62no—
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—61no—
155C2×C3×D3—61no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C32×C4—61no—
159C32×C4—61no—
160C3⋊Dic3—21no—
161C3⋊Dic3—21no—
162C3⋊Dic3—32no—
163C3⋊Dic3—32no—
164C3⋊Dic3—32no—
165C3⋊Dic3—62no—
166C3⋊Dic3—62no—
167C3⋊Dic3—62no—
168C3⋊Dic3—122no—
169C3×Dic3—22no—
170C3×Dic3—22no—
171C3×Dic3—21no—
172C3×Dic3—21no—
173C3×Dic3—21no—
174C3×Dic3—21no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181#124—271noSylow
182C33#12411yescharacteristic, complemented, Sylow
183C23×C3—31no—
184C23×C3—31no—
185C23×C3—61no—
186C23×C3—91no—
187C22×D3—31no—
188C22×D3—31no—
189C22×D3—61no—
190C22×D3D3211yescharacteristic
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199#67—182no—
200#67—182no—
201#67—182no—
202#67—184no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—31no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—181no—
215C2×Dic3—361no—
216C2×Dic3D3211yescharacteristic
217C2×Dic3D3211yescharacteristic
218Dic6—182no—
219Dic6—182no—
220Dic6—182no—
221Dic6—184no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—121no—
235C2×C32#23511yescharacteristic
236C2×C32#23511yescharacteristic
237C2×C32#23511yescharacteristic
238C2×C32D3×D411yescharacteristic
239C2×C32D3×D411yescharacteristic
240C2×C32D3×D411yescharacteristic
241C2×C32#23711yescharacteristic
242C2×C32#23711yescharacteristic
243C2×C32#23711yescharacteristic
244C3⋊D3—41no—
245C3×D3—42no—
246C3×D3—41no—
247C3×D3—41no—
248C3×D3—122no—
249C3×D3—121no—
250C3×D3—121no—
251C2×Q8—271no—
252C2×D4—271no—
253C22⋊C4—91no—
254C22⋊C4—91no—
255C22⋊C4—91no—
256C22⋊C4—271no—
257C42—271no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—41no—
262C22×C3—62no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—121no—
270C22×C3—121no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3C2×D3211yescharacteristic
275C22×C3C2×D3211yescharacteristic
276C22×C3C2×D3211yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—62no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—61no—
287C2×D3—121no—
288C2×D3—121no—
289C3×C4—62no—
290C3×C4—62no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—181no—
296C3×C4—181no—
297Dic3—21no—
298Dic3—21no—
299Dic3—62no—
300Dic3—62no—
301Dic3—61no—
302Dic3—61no—
303Dic3—61no—
304Dic3—61no—
305Dic3—92no—
306Dic3—92no—
307Dic3—92no—
308Dic3—182no—
309Dic3—182no—
310Dic3—182no—
311Dic3—362no—
312C32—21no—
313C32—21no—
314C32—21no—
315C32—41no—
316C32#88511yescharacteristic, complemented
317C32#88511yescharacteristic, complemented
318C32#88511yescharacteristic, complemented
319C23—31no—
320C23—91no—
321Q8—542no—
322D4—542no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—271no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—122no—
341C2×C3—121no—
342C2×C3—121no—
343C2×C3—361no—
344C2×C3#379511yescharacteristic
345C2×C3#379511yescharacteristic
346C2×C3#379811yescharacteristic
347C2×C3#380211yescharacteristic
348C2×C3#380411yescharacteristic
349C2×C3#380411yescharacteristic
350C2×C3#380511yescharacteristic
351C2×C3#381111yescharacteristic
352C2×C3#381111yescharacteristic
353D3—41no—
354D3—122no—
355D3—121no—
356D3—121no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22—181no—
361C22—181no—
362C22—181no—
363C22D3311yescharacteristic
364C4—61no—
365C4—61no—
366C4—181no—
367C4—181no—
368C4—272no—
369C3—21no—
370C3—21no—
371C3—21no—
372C3—41no—
373C3#6391311yescharacteristic, complemented
374C3#6391511yescharacteristic, complemented
375C3#6395311yescharacteristic, complemented
376C2—121no—
377C2—361no—
378C2#8895411yescharacteristic
379C2#8896011yescharacteristic
380C2#8896111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6290C2
hypocenterC33#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6293C2semidirect
3. #88814C2semidirect
4. Dic3×C3⋊Dic3C2semidirect
5. C3×#3722C2semidirect
6. C3×(C2×D3)⋊Dic3C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
36. #3722D3semidirect
39. (C2×D3)⋊Dic3D3semidirect
40. (C2×D3)⋊Dic3D3semidirect
182. C33#124semidirect
316. C32#885semidirect
317. C32#885semidirect
318. C32#885semidirect
373. C3#63913semidirect
374. C3#63915semidirect
375. C3#63953semidirect

Maximal subgroups

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

subgroupindex
C2×#62932
C2×#62902
#888142
Dic3×C3⋊Dic32
C3×#37222
C3×(C2×D3)⋊Dic32
#639533
#639153
#639133