Finite Group Databasebeta

← #11722989

Subgroups — #11722989

Order 864.

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

normal subgroups only

This group has 442 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×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#625C211yesmaximal, characteristic
3#88815C211yesmaximal, characteristic, complemented
4Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
8#63969—31nomaximal
9#63962—31nomaximal
10#63891—31nomaximal
11C22×C32×D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18D3×C3⋊Dic3—22no—
19C3×#625—22no—
20C22×C3⋊Dic3—31no—
21C2×C3×#67—31no—
22C2×C3×#67—31no—
23C2×#625D311yescharacteristic
24C2×D3×Dic3—31no—
25C2×D3×Dic3—31no—
26C2×D3×Dic3—61no—
27#3757—31no—
28C4⋊(C3⋊Dic3)—31no—
29C3×#217—31no—
30C3×#217—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—31no—
33#3724—31no—
34Dic3⋊Dic3—31no—
35Dic3⋊Dic3—31no—
36Dic3⋊Dic3—31no—
37Dic3⋊Dic3—31no—
38Dic3⋊Dic3—61no—
39(C2×D3)⋊Dic3—31no—
40(C2×D3)⋊Dic3—31no—
41(C2×D3)⋊Dic3—61no—
42(C2×D3)⋊Dic3D311yescharacteristic, complemented
43(C2×D3)⋊Dic3D311yescharacteristic, complemented
44C22×C33C2311yescharacteristic
45C2×C32×D3—21no—
46C2×C32×D3—21no—
47C2×C32×D3—21no—
48C2×C32×D3—21no—
49C2×C32×D3—21no—
50C2×C32×D3D412yes—
51C3⋊(C3⋊Dic3)—21no—
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—21no—
55C32×Dic3—21no—
56#935—91no—
57#886—91no—
58#883—91no—
59C23×C32—31no—
60C23×C32—31no—
61C22×C3×D3—21no—
62C22×C3×D3—21no—
63C22×C3×D3C2×D311yescharacteristic
64C22×C3×D3C2×D311yescharacteristic
65C22×C3×D3C2×D311yescharacteristic
66C22×C3×D3C2×D311yescharacteristic
67C2×C32×C4—31no—
68C2×C3⋊Dic3—31no—
69C2×C3⋊Dic3—31no—
70C2×C3⋊Dic3—31no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—61no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—62no—
75C2×C3⋊Dic3—121no—
76C2×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊Dic3C2×D311yescharacteristic
79C3×#67—62no—
80C3×#67—62no—
81C2×C3×Dic3—21no—
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—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3C2×D311yescharacteristic
92C2×C3×Dic3C2×D311yescharacteristic
93#625—22no—
94D3×Dic3—64no—
95D3×Dic3—62no—
96D3×Dic3—62no—
97C2×C33C2×D411yescharacteristic
98C2×C33#4111yescharacteristic
99C2×C33#4111yescharacteristic
100C32×D3—22no—
101C32×D3—41no—
102C2×C3×D4—91no—
103C2×#67—31no—
104C2×#67—31no—
105C22×Dic3—91no—
106C22×Dic3—91no—
107C22×Dic3—181no—
108C2×C4×D3—91no—
109C3×C22⋊C4—91no—
110C3×C22⋊C4—91no—
111#217—31no—
112#217—31no—
113#217—91no—
114#217—91no—
115#217—181no—
116(C2×D3)⋊C4—31no—
117(C2×D3)⋊C4—31no—
118(C2×D3)⋊C4—91no—
119C4⋊Dic3—91no—
120C4⋊Dic3—91no—
121C4⋊Dic3—91no—
122C4⋊Dic3—181no—
123Dic3⋊C4—91no—
124Dic3⋊C4—91no—
125Dic3⋊C4—91no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—32no—
130C22×C32—41no—
131C22×C32—61no—
132C22×C32—61no—
133C22×C32—61no—
134C22×C32—61no—
135C22×C32—61no—
136C22×C32C22×D311yescharacteristic
137C22×C32C22×D311yescharacteristic
138C22×C32C22×D311yescharacteristic
139C2×C3×D3—22no—
140C2×C3×D3—22no—
141C2×C3×D3—21no—
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3—22no—
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—22no—
153C2×C3×D3—41no—
154C2×C3×D3—41no—
155C2×C3×D3D1212yes—
156C2×C3×D3#6712yes—
157C32×C4—61no—
158C3⋊Dic3—21no—
159C3⋊Dic3—21no—
160C3⋊Dic3—21no—
161C3⋊Dic3—61no—
162C3⋊Dic3—62no—
163C3⋊Dic3—61no—
164C3⋊Dic3—61no—
165C3⋊Dic3—121no—
166C3⋊Dic3—121no—
167C3⋊Dic3—122no—
168C3×Dic3—22no—
169C3×Dic3—21no—
170C3×Dic3—21no—
171C3×Dic3—62no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180#123—271noSylow
181C33#12311yescharacteristic, complemented, Sylow
182C23×C3—31no—
183C23×C3—31no—
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—61no—
187C23×C3—61no—
188C22×D3D3211yescharacteristic
189C22×D3D3211yescharacteristic
190C3×D4—182no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197#67—62no—
198#67—62no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—91no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—182no—
215C2×Dic3—182no—
216C2×Dic3—184no—
217C2×Dic3—361no—
218C2×Dic3D3211yescharacteristic
219C4×D3—182no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—62no—
233C2×C32—121no—
234C2×C32C2×D1211yescharacteristic
235C2×C32#23511yescharacteristic
236C2×C32D3×D411yescharacteristic
237C2×C32#23711yescharacteristic
238C2×C32#23711yescharacteristic
239C2×C32#23711yescharacteristic
240C2×C32#23711yescharacteristic
241C2×C32#23711yescharacteristic
242C2×C32C2×#6711yescharacteristic
243C3×D3—24no—
244C3×D3—22no—
245C3×D3—22no—
246C3×D3—42no—
247C3×D3—41no—
248C3×D3—41no—
249C2×D4—91no—
250C22×C4—271no—
251C4⋊C4—271no—
252C4⋊C4—271no—
253C22⋊C4—91no—
254C22⋊C4—91no—
255C22⋊C4—271no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—32no—
260C22×C3—32no—
261C22×C3—41no—
262C22×C3—62no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3C2×D3211yescharacteristic
279C22×C3C2×D3211yescharacteristic
280C22×C3C2×D3211yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3#62612yes—
287C3×C4—62no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—181no—
291C3×C4—181no—
292C3×C4—181no—
293Dic3—21no—
294Dic3—62no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—61no—
300Dic3—61no—
301Dic3—121no—
302Dic3—121no—
303Dic3—182no—
304Dic3—181no—
305Dic3—181no—
306Dic3—181no—
307Dic3—361no—
308Dic3—361no—
309Dic3—362no—
310C32—21no—
311C32—21no—
312C32—21no—
313C32—41no—
314C32#88311yescharacteristic, complemented
315C32#88611yescharacteristic, complemented
316C32#93511yescharacteristic, complemented
317C23—31no—
318C23—31no—
319D4—182no—
320C2×C4—31no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—271no—
325C2×C4—542no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—64no—
339C2×C3—62no—
340C2×C3—62no—
341C2×C3—122no—
342C2×C3—121no—
343C2×C3—121no—
344C2×C3#379511yescharacteristic
345C2×C3D3×D1211yescharacteristic
346C2×C3#380411yescharacteristic
347C2×C3#380511yescharacteristic
348C2×C3#380511yescharacteristic
349C2×C3#380511yescharacteristic
350C2×C3#380511yescharacteristic
351C2×C3C2×#62611yescharacteristic
352C2×C3D3×#6711yescharacteristic
353D3—22no—
354D3—41no—
355C22—32no—
356C22—61no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22—61no—
361C22D3311yescharacteristic
362C4—61no—
363C4—181no—
364C4—181no—
365C4—181no—
366C4—541no—
367C3—21no—
368C3—21no—
369C3—21no—
370C3—41no—
371C3#6389111yescharacteristic, complemented
372C3#6396211yescharacteristic, complemented
373C3#6396911yescharacteristic, complemented
374C2—62no—
375C2—121no—
376C2D3×#62611yescharacteristic
377C2#8896111yescharacteristic
378C2#8896311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×C3×#625C2
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊Dic3C2semidirect
3. #88815C2semidirect
4. Dic3⋊(C3⋊Dic3)C2semidirect
5. (C2×D3)⋊(C3⋊Dic3)C2semidirect
6. C3×(C2×D3)⋊Dic3C2semidirect
7. C3×(C2×D3)⋊Dic3C2semidirect
42. (C2×D3)⋊Dic3D3semidirect
43. (C2×D3)⋊Dic3D3semidirect
181. C33#123semidirect
314. C32#883semidirect
315. C32#886semidirect
316. C32#935semidirect
371. C3#63891semidirect
372. C3#63962semidirect
373. C3#63969semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊Dic32
C2×C3×#6252
#888152
Dic3⋊(C3⋊Dic3)2
(C2×D3)⋊(C3⋊Dic3)2
C3×(C2×D3)⋊Dic32
#639693
#639623
#638913