Finite Group Databasebeta

← #78681

Subgroups — #78681

Order 384.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5833C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3C3×#408C211yesmaximal, characteristic, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5134C211yesmaximal, characteristic, complemented
6#5116C211yesmaximal, characteristic, complemented
7#5114C211yesmaximal, characteristic, complemented
8#4739C211yesmaximal, characteristic, complemented
9#4738C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4733C211yesmaximal, characteristic, complemented
12#4733C211yesmaximal, characteristic, complemented
13#4723C211yesmaximal, characteristic, complemented
14#4722C211yesmaximal, characteristic, complemented
15#4721C211yesmaximal, characteristic, complemented
16#3201—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C22×Dic6C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic
21C22×C3×C8C2211yescharacteristic
22#946—21no—
23#944—21no—
24#939—21no—
25#939—21no—
26#938—21no—
27#936—21no—
28#935—21no—
29#935—21no—
30#930C2211yescharacteristic
31#930C2211yescharacteristic
32#926C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34#924C2211yescharacteristic
35#924C2211yescharacteristic
36#924C2211yescharacteristic
37#924C2211yescharacteristic
38#924C2211yescharacteristic
39#924C2211yescharacteristic
40C2×Dic12C2211yescharacteristic
41C2×Dic12C2211yescharacteristic
42C2×#204C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#818C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58#816C2211yescharacteristic
59#816C2211yescharacteristic
60#537—31no—
61#536—31no—
62#485—31no—
63#484—31no—
64#483—31no—
65#482—31no—
66#470—31no—
67#468—31no—
68#467—31no—
69#465—31no—
70#452—31no—
71#452—31no—
72#451—31no—
73#448—31no—
74#408D311yescharacteristic, complemented
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8D411yescharacteristic
78C2×C3×D4D411yescharacteristic
79C2×C3×D4D411yescharacteristic
80C2×C3×D4D411yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89#235—41no—
90#235—41no—
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×Dic6—21no—
94C2×Dic6—41no—
95C2×Dic6—41no—
96C2×Dic6C2311yescharacteristic
97C2×Dic6C2311yescharacteristic
98C2×Dic6C2311yescharacteristic
99C3×M16—41no—
100C2×C3×C8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126Dic12—41no—
127Dic12—41no—
128#204—41no—
129#204—41no—
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C22×Q8—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×M16C2×D311yescharacteristic
138C22×C8C2×D311yescharacteristic
139C4⋊Q8—61no—
140#124—61no—
141#123—61no—
142#123—61no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145C22⋊Q8—31no—
146C22⋊Q8—31no—
147C22⋊Q8—31no—
148C22⋊Q8—31no—
149C22⋊Q8—61no—
150#121—31no—
151#121—31no—
152#121—61no—
153#121—61no—
154#121—61no—
155#117—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103—31no—
167#102—31no—
168#102—31no—
169C22⋊C8C2×D311yescharacteristic
170C22⋊C8C2×D311yescharacteristic
171C22⋊C8C2×D311yescharacteristic
172C22⋊C8C2×D311yescharacteristic
173C23×C3C2×D411yescharacteristic
174C23×C3C2×D411yescharacteristic
175C23×C3C2×D411yescharacteristic
176C22×D3—21no—
177C3×Q8—41no—
178C3×D4—41no—
179C3×D4—41no—
180C3×D4—41no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—42no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—42no—
202C2×Dic3—81no—
203D12—21no—
204D12—21no—
205C4×D3—42no—
206Dic6—21no—
207Dic6—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—21no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3×C8—21no—
216C3×C8—21no—
217#41—41no—
218#41—41no—
219#41—121no—
220#41—121no—
221C2×Q8—31no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—61no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×Q8D1211yescharacteristic
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4D1211yescharacteristic
233C2×D4D1211yescharacteristic
234C2×D4D1211yescharacteristic
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241Q16—121no—
242Q16—121no—
243SD16—121no—
244SD16—121no—
245M16—41no—
246C2×C8—41no—
247C2×C8C22×D311yescharacteristic
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C2×C8C22×D311yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3C22×D411yescharacteristic
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4H(5,𝔽2)11yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—41no—
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—61no—
298C23C2×D1211yescharacteristic
299C23C2×D1211yescharacteristic
300C23C2×D1211yescharacteristic
301Q8—41no—
302Q8—61no—
303Q8—61no—
304Q8—61no—
305Q8—61no—
306Q8—61no—
307Q8—61no—
308Q8—122no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—61no—
313D4—61no—
314D4—122no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4C2×D1211yescharacteristic
334C2×C4C2×D1211yescharacteristic
335C2×C4C2×D1211yescharacteristic
336C2×C4C23×D311yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3#53411yescharacteristic
345C2×C3#57711yescharacteristic
346C2×C3#57811yescharacteristic
347D3—81no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C22×D1211yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—121no—
365C4—121no—
366C4—121no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100411yescharacteristic
373C3#320111yescharacteristic, complemented, Sylow
374C2—41no—
375C2—41no—
376C2—41no—
377C2—241no—
378C2#549311yescharacteristic
379C2#575111yescharacteristic
380C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
prime radical#5833C2
hypercenter#408D3
hypocenterC3#3201
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C3×#408C2semidirect
4. #5137C2semidirect
5. #5134C2semidirect
6. #5116C2semidirect
7. #5114C2semidirect
8. #4739C2semidirect
9. #4738C2semidirect
10. #4737C2semidirect
11. #4733C2semidirect
12. #4733C2semidirect
13. #4723C2semidirect
14. #4722C2semidirect
15. #4721C2semidirect
74. #408D3semidirect
373. C3#3201semidirect

Maximal subgroups

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

subgroupindex
#58332
#58252
C3×#4082
#51372
#51342
#51162
#51142
#47392
#47382
#47372
#47332
#47232
#47222
#47212
#32013