Finite Group Databasebeta

← #11723001

Subgroups — #11723001

Order 864.

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

normal subgroups only

This group has 418 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×C3×#627C211yesmaximal, characteristic, complemented
2C2×C3×#626C211yesmaximal, characteristic
3#88814C211yesmaximal, characteristic, complemented
4#88803C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63959—31nomaximal
9#63907—31nomaximal
10#63863—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
18C3×#627—22no—
19C3×#626—22no—
20C2×C3×#67—31no—
21C2×C3×D12—31no—
22C2×C3×Dic6—31no—
23C2×C3×Dic6—31no—
24C2×#627D311yescharacteristic, complemented
25C2×#626D311yescharacteristic
26#3757—31no—
27#3752—31no—
28C4×C3⋊Dic3—31no—
29C3×#217—31no—
30C3×(C2×D3)⋊C4—31no—
31#3722—31no—
32(C2×D3)⋊Dic3—31no—
33(C2×D3)⋊Dic3—31no—
34(C2×D3)⋊Dic3—31no—
35(C2×D3)⋊Dic3—31no—
36(C2×D3)⋊Dic3—31no—
37(C2×D3)⋊Dic3—31no—
38(C2×D3)⋊Dic3—61no—
39(C2×D3)⋊Dic3—61no—
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)—21no—
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—21no—
54C32×Dic3—21no—
55C32×Dic3D412yes—
56#947—91no—
57#936—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—
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—121no—
76C2×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊Dic3C2×D311yescharacteristic
78C3×#67—62no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—31no—
83C2×C3×Dic3—31no—
84C2×C3×Dic3—31no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3C2×D311yescharacteristic
88C2×C3×Dic3C2×D311yescharacteristic
89C2×C3×Dic3C2×D311yescharacteristic
90C2×C3×Dic3C2×D311yescharacteristic
91C3×D12—62no—
92C3×Dic6—62no—
93C3×Dic6—62no—
94#627—22no—
95#626—22no—
96C2×C33C2×D411yescharacteristic
97C2×C33#4111yescharacteristic
98C2×C33#4111yescharacteristic
99C3×C3⋊D3—41no—
100C32×D3—41no—
101C2×C3×Q8—91no—
102C2×C3×D4—91no—
103C2×#67—31no—
104C2×D12—31no—
105C2×Dic6—31no—
106C2×Dic6—31no—
107C3×C22⋊C4—91no—
108#217—31no—
109#217—91no—
110#217—91no—
111#217—91no—
112#217—91no—
113#217—181no—
114(C2×D3)⋊C4—31no—
115(C2×D3)⋊C4—91no—
116(C2×D3)⋊C4—91no—
117(C2×D3)⋊C4—91no—
118(C2×D3)⋊C4—91no—
119(C2×D3)⋊C4—91no—
120(C2×D3)⋊C4—181no—
121C4×Dic3—91no—
122C4×Dic3—91no—
123C4×Dic3—91no—
124C4×Dic3—181no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—41no—
129C22×C32—61no—
130C22×C32—61no—
131C22×C32—61no—
132C22×C32C22×D311yescharacteristic
133C22×C32C22×D311yescharacteristic
134C22×C32C22×D311yescharacteristic
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3×D3—22no—
139C2×C3×D3—21no—
140C2×C3×D3—21no—
141C2×C3×D3—21no—
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3—41no—
146C2×C3×D3—41no—
147C2×C3×D3—62no—
148C2×C3×D3—61no—
149C2×C3×D3—61no—
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C32×C4—32no—
157C32×C4—61no—
158C3⋊Dic3—21no—
159C3⋊Dic3—21no—
160C3⋊Dic3—61no—
161C3⋊Dic3—62no—
162C3⋊Dic3—61no—
163C3⋊Dic3—61no—
164C3⋊Dic3—121no—
165C3⋊Dic3—121no—
166C3⋊Dic3—122no—
167C3×Dic3—21no—
168C3×Dic3—21no—
169C3×Dic3—22no—
170C3×Dic3—41no—
171C3×Dic3—62no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—121no—
177C3×Dic3#6712yes—
178C3×Dic3#6712yes—
179#124—271noSylow
180C33#12411yescharacteristic, complemented, Sylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3—91no—
185C22×D3—31no—
186C22×D3—31no—
187C22×D3—61no—
188C22×D3D3211yescharacteristic
189C3×Q8—182no—
190C3×D4—182no—
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—62no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C2×Dic3—91no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—361no—
213C2×Dic3D3211yescharacteristic
214C2×Dic3D3211yescharacteristic
215D12—62no—
216Dic6—62no—
217Dic6—62no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—121no—
231C2×C32#23511yescharacteristic
232C2×C32D3×D411yescharacteristic
233C2×C32#23711yescharacteristic
234C2×C32#23711yescharacteristic
235C2×C32#23711yescharacteristic
236C2×C32#23911yescharacteristic
237C2×C32#23911yescharacteristic
238C2×C32C2×#6711yescharacteristic
239C2×C32C2×#6711yescharacteristic
240C3⋊D3—41no—
241C3×D3—42no—
242C3×D3—41no—
243C3×D3—41no—
244C3×D3—122no—
245C3×D3—121no—
246C3×D3—121no—
247C2×Q8—91no—
248C2×D4—91no—
249C22⋊C4—91no—
250C22⋊C4—271no—
251C22⋊C4—271no—
252C22⋊C4—271no—
253C42—271no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—41no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—121no—
266C22×C3—121no—
267C22×C3—181no—
268C22×C3—181no—
269C22×C3—181no—
270C22×C3C2×D3211yescharacteristic
271C22×C3C2×D3211yescharacteristic
272C22×C3C2×D3211yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—62no—
277C2×D3—61no—
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—61no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—121no—
284C2×D3—121no—
285C3×C4—32no—
286C3×C4—32no—
287C3×C4—62no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—121no—
291C3×C4—181no—
292C3×C4—181no—
293Dic3—21no—
294Dic3—62no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—121no—
300Dic3—182no—
301Dic3—181no—
302Dic3—181no—
303Dic3—181no—
304Dic3—361no—
305Dic3—361no—
306Dic3—362no—
307Dic3#62512yes—
308C32—21no—
309C32—21no—
310C32—21no—
311C32—41no—
312C32#88511yescharacteristic, complemented
313C32#93611yescharacteristic, complemented
314C32#94711yescharacteristic, complemented
315C23—31no—
316C23—91no—
317Q8—182no—
318D4—182no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—271no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—122no—
337C2×C3—121no—
338C2×C3—121no—
339C2×C3—361no—
340C2×C3#379611yescharacteristic
341C2×C3#379611yescharacteristic
342C2×C3#379611yescharacteristic
343C2×C3#379911yescharacteristic
344C2×C3#380411yescharacteristic
345C2×C3#380511yescharacteristic
346C2×C3C2×#62511yescharacteristic
347C2×C3D3×#6711yescharacteristic
348C2×C3D3×#6711yescharacteristic
349D3—41no—
350D3—122no—
351D3—121no—
352D3—121no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22—181no—
357C22—181no—
358C22—181no—
359C22D3311yescharacteristic
360C4—32no—
361C4—61no—
362C4—181no—
363C4—181no—
364C4—541no—
365C3—21no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3#6386311yescharacteristic, complemented
370C3#6390711yescharacteristic, complemented
371C3#6395911yescharacteristic, complemented
372C2—121no—
373C2—361no—
374C2D3×#62511yescharacteristic
375C2#8896211yescharacteristic
376C2#8896411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×C3×#626C2
hypocenterC33#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#627C2semidirect
3. #88814C2semidirect
4. #88803C2semidirect
5. (C2×D3)⋊(C3⋊Dic3)C2semidirect
6. Dic3×C3⋊Dic3C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
24. C2×#627D3semidirect
40. (C2×D3)⋊Dic3D3semidirect
180. C33#124semidirect
312. C32#885semidirect
313. C32#936semidirect
314. C32#947semidirect
369. C3#63863semidirect
370. C3#63907semidirect
371. C3#63959semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6272
C2×C3×#6262
#888142
#888032
(C2×D3)⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
C3×(C2×D3)⋊Dic32
#639593
#639073
#638633