Finite Group Databasebeta

← #11723021

Subgroups — #11723021

Order 864.

374 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#88815C211yesmaximal, characteristic, complemented
2#88804C211yesmaximal, characteristic
3#88804C211yesmaximal, characteristic
4#88804C211yesmaximal, characteristic
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic32C211yesmaximal, characteristic, complemented
8#63868—31nomaximal
9#63868—31nomaximal
10#63848—31nomaximal
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#3752—31no—
19#3752—31no—
20#3752—31no—
21C3×C4⋊Dic3—31no—
22C3×Dic3⋊C4—31no—
23C3×Dic3⋊C4—31no—
24C3×Dic3⋊C4—31no—
25C3×C4×Dic3—31no—
26C3×C4×Dic3—31no—
27#3724—31no—
28#3724D311yescharacteristic, complemented
29Dic3⋊Dic3—31no—
30Dic3⋊Dic3—31no—
31Dic3⋊Dic3D311yescharacteristic, complemented
32#3722—31no—
33#3722—31no—
34#3722—31no—
35#3722—31no—
36#3722—31no—
37#3722—31no—
38#3722—61no—
39#3722—61no—
40#3722—61no—
41Dic32D311yescharacteristic, complemented
42C22×C33C2311yescharacteristic
43C2×C3⋊(C3⋊D3)—21no—
44C2×C3⋊(C3⋊D3)—21no—
45C2×C3⋊(C3⋊D3)—21no—
46C3×C3⋊Dic3—21no—
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3—21no—
49C32×Dic3—21no—
50C32×Dic3—21no—
51C32×Dic3—21no—
52#898—91no—
53#898—91no—
54#876—91no—
55C22×C3⋊D3—31no—
56C22×C3⋊D3—31no—
57C22×C3⋊D3—31no—
58C22×C3⋊D3—61no—
59C22×C3⋊D3—61no—
60C22×C3⋊D3—61no—
61C22×C3⋊D3—121no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊Dic3C2×D311yescharacteristic
68C2×C3×Dic3—21no—
69C2×C3×Dic3—21no—
70C2×C3×Dic3—21no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3C2×D311yescharacteristic
81C2×C3×Dic3C2×D311yescharacteristic
82C2×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C2×D311yescharacteristic
86C2×C33#4111yescharacteristic
87C2×C33#4111yescharacteristic
88C2×C33#4111yescharacteristic
89C3⋊(C3⋊D3)—41no—
90C3×C4⋊C4—91no—
91C3×C4⋊C4—91no—
92C3×C42—91no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—91no—
95(C2×D3)⋊C4—91no—
96(C2×D3)⋊C4—91no—
97(C2×D3)⋊C4—91no—
98(C2×D3)⋊C4—91no—
99(C2×D3)⋊C4—91no—
100(C2×D3)⋊C4—91no—
101(C2×D3)⋊C4—91no—
102(C2×D3)⋊C4—181no—
103(C2×D3)⋊C4—181no—
104(C2×D3)⋊C4—181no—
105C4⋊Dic3—31no—
106C4⋊Dic3—91no—
107Dic3⋊C4—31no—
108Dic3⋊C4—31no—
109Dic3⋊C4—31no—
110Dic3⋊C4—91no—
111Dic3⋊C4—91no—
112C4×Dic3—31no—
113C4×Dic3—31no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—41no—
118C22×C32C22×D311yescharacteristic
119C22×C32C22×D311yescharacteristic
120C22×C32C22×D311yescharacteristic
121C2×C3⋊D3—61no—
122C2×C3⋊D3—61no—
123C2×C3⋊D3—61no—
124C2×C3⋊D3—62no—
125C2×C3⋊D3—62no—
126C2×C3⋊D3—62no—
127C2×C3⋊D3—61no—
128C2×C3⋊D3—61no—
129C2×C3⋊D3—61no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—61no—
132C2×C3⋊D3—61no—
133C2×C3⋊D3—121no—
134C2×C3⋊D3—121no—
135C2×C3⋊D3—121no—
136C2×C3⋊D3—121no—
137C2×C3⋊D3—121no—
138C2×C3⋊D3—121no—
139C2×C3⋊D3—122no—
140C2×C3⋊D3—122no—
141C2×C3⋊D3—122no—
142C32×C4—61no—
143C32×C4—61no—
144C32×C4—61no—
145C3⋊Dic3—21no—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3—21no—
150C3×Dic3—21no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—41no—
155C3×Dic3—41no—
156C3×Dic3—41no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—121no—
164C3×Dic3—121no—
165C3×Dic3—121no—
166#126—271noSylow
167C33#12611yescharacteristic, complemented, Sylow
168C22×D3—91no—
169C22×D3—91no—
170C22×D3—91no—
171C22×D3—181no—
172C22×D3—181no—
173C22×D3—181no—
174C22×D3—361no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4—91no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3D3211yescharacteristic
197C2×Dic3D3211yescharacteristic
198C2×Dic3D3211yescharacteristic
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32#23511yescharacteristic
212C2×C32#23511yescharacteristic
213C2×C32#23511yescharacteristic
214C2×C32#23511yescharacteristic
215C2×C32#23511yescharacteristic
216C2×C32#23711yescharacteristic
217C2×C32#23711yescharacteristic
218C2×C32#23911yescharacteristic
219C2×C32#23911yescharacteristic
220C3⋊D3—122no—
221C3⋊D3—122no—
222C3⋊D3—122no—
223C3⋊D3—124no—
224C3⋊D3—121no—
225C3⋊D3—121no—
226C3⋊D3—121no—
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—271no—
230C22⋊C4—271no—
231C22⋊C4—271no—
232C22⋊C4—271no—
233C42—91no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3C2×D3211yescharacteristic
239C22×C3C2×D3211yescharacteristic
240C22×C3C2×D3211yescharacteristic
241C2×D3—182no—
242C2×D3—182no—
243C2×D3—182no—
244C2×D3—181no—
245C2×D3—181no—
246C2×D3—181no—
247C2×D3—181no—
248C2×D3—181no—
249C2×D3—181no—
250C2×D3—181no—
251C2×D3—181no—
252C2×D3—181no—
253C2×D3—361no—
254C2×D3—361no—
255C2×D3—361no—
256C2×D3—361no—
257C2×D3—361no—
258C2×D3—361no—
259C2×D3—362no—
260C2×D3—362no—
261C2×D3—362no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—121no—
271C3×C4—181no—
272C3×C4—181no—
273C3×C4—181no—
274Dic3—21no—
275Dic3—21no—
276Dic3—21no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—121no—
284Dic3—121no—
285Dic3—121no—
286C32—21no—
287C32—21no—
288C32—21no—
289C32—41no—
290C32#87611yescharacteristic, complemented
291C32#89811yescharacteristic, complemented
292C32#89811yescharacteristic, complemented
293C23—271no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3#379611yescharacteristic
313C2×C3#379811yescharacteristic
314C2×C3#379811yescharacteristic
315C2×C3#379911yescharacteristic
316C2×C3#379911yescharacteristic
317C2×C3#379911yescharacteristic
318C2×C3#380411yescharacteristic
319C2×C3#380411yescharacteristic
320C2×C3#380411yescharacteristic
321D3—362no—
322D3—362no—
323D3—362no—
324D3—364no—
325D3—361no—
326D3—361no—
327D3—361no—
328C22—541no—
329C22—541no—
330C22—541no—
331C22D3311yescharacteristic
332C4—61no—
333C4—61no—
334C4—61no—
335C4—181no—
336C4—181no—
337C4—181no—
338C3—21no—
339C3—21no—
340C3—21no—
341C3—41no—
342C3#6384811yescharacteristic, complemented
343C3#6386811yescharacteristic, complemented
344C3#6386811yescharacteristic, complemented
345C2—1081no—
346C2#8895911yescharacteristic
347C2#8896211yescharacteristic
348C2#8896511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C3⋊(C3⋊D3)C22
hypocenterC33#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #88815C2semidirect
5. C3×#3724C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
7. C3×Dic32C2semidirect
28. #3724D3semidirect
31. Dic3⋊Dic3D3semidirect
41. Dic32D3semidirect
167. C33#126semidirect
290. C32#876semidirect
291. C32#898semidirect
292. C32#898semidirect
342. C3#63848semidirect
343. C3#63868semidirect
344. C3#63868semidirect

Maximal subgroups

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

subgroupindex
#888152
#888042
C3×#37242
C3×Dic3⋊Dic32
C3×Dic322
#638683
#638483