Finite Group Databasebeta

← #83408

Subgroups — #83408

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
1C3×#536C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5227C211yesmaximal, characteristic, complemented
6#5052C211yesmaximal, characteristic, complemented
7#5051C211yesmaximal, characteristic, complemented
8#5049C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#5047C211yesmaximal, characteristic, complemented
11#5047C211yesmaximal, characteristic, complemented
12#5044C211yesmaximal, characteristic, complemented
13#5043C211yesmaximal, characteristic, complemented
14#5003C211yesmaximal, characteristic, complemented
15#4996C211yesmaximal, characteristic, complemented
16#3201—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C3×C4⋊Q8—21no—
21C3×#124—21no—
22C3×#123—21no—
23C3×#123—21no—
24C3×C22⋊Q8C2211yescharacteristic
25C3×C22⋊Q8C2211yescharacteristic
26C3×C22⋊Q8C2211yescharacteristic
27C3×C22⋊Q8C2211yescharacteristic
28C3×#117C2211yescharacteristic
29#946—21no—
30C2×#216C2211yescharacteristic
31C2×#216C2211yescharacteristic
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic
34#939—21no—
35#939—21no—
36#938—21no—
37#930C2211yescharacteristic
38#930C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40#924C2211yescharacteristic
41#924C2211yescharacteristic
42C2×#208C2211yescharacteristic
43C22×C3⋊C8C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#830C2211yescharacteristic
49#830C2211yescharacteristic
50#830C2211yescharacteristic
51#830C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#537—31no—
61#536D311yescharacteristic, complemented
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#408—31no—
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8—21no—
78C2×C3×Q8—41no—
79C2×C3×Q8—41no—
80C2×C3×Q8D411yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4D411yescharacteristic
84C2×C3×D4D411yescharacteristic
85C2×C3×D4D411yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C22×Dic3—21no—
94#235—41no—
95#235—41no—
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×Dic6C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C42—21no—
111#217—21no—
112#217—21no—
113#216—41no—
114#216—41no—
115#215—41no—
116#215—41no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124#208—41no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C22×Q8C2×D311yescharacteristic
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×M16—31no—
138C22×C8—31no—
139C4⋊Q8—21no—
140#124—21no—
141#123—21no—
142#123—21no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145C22⋊Q8—61no—
146C22⋊Q8C2×D311yescharacteristic
147C22⋊Q8C2×D311yescharacteristic
148C22⋊Q8C2×D311yescharacteristic
149C22⋊Q8C2×D311yescharacteristic
150#121—31no—
151#121—31no—
152#121—61no—
153#121—61no—
154#121—61no—
155#117C2×D311yescharacteristic
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⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3C2×D411yescharacteristic
174C23×C3C2×D411yescharacteristic
175C23×C3C2×D411yescharacteristic
176C22×D3—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—41no—
182C3×Q8—42no—
183C3×D4—41no—
184C3×D4—41no—
185C3×D4—41no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—81no—
203#67—81no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—81no—
208D12—21no—
209D12—21no—
210C4×D3—42no—
211Dic6—21no—
212Dic6—21no—
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—21no—
222C2×Q8—31no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8#6711yescharacteristic
226C2×Q8C22×D311yescharacteristic
227C2×Q8C22×D311yescharacteristic
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4#6711yescharacteristic
233C2×D4#6711yescharacteristic
234C2×D4#6711yescharacteristic
235C22×C4—21no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241Q16—121no—
242Q16—121no—
243SD16—121no—
244SD16—121no—
245M16—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—31no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—21no—
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×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4H(5,𝔽2)11yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—61no—
298C23C2×#6711yescharacteristic
299C23C2×#6711yescharacteristic
300C23C2×#6711yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—41no—
306Q8—42no—
307Q8—61no—
308Q8—61no—
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—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4C2×#6711yescharacteristic
334C2×C4C2×#6711yescharacteristic
335C2×C4C2×#6711yescharacteristic
336C2×C4C23×D311yescharacteristic
337C8—61no—
338C8—61no—
339C8—61no—
340C8—61no—
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×#6711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100411yescharacteristic
373C3#320111yescharacteristic, complemented, Sylow
374C2—41no—
375C2—41no—
376C2—41no—
377C2—241no—
378C2#580411yescharacteristic
379C2#583711yescharacteristic
380C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radical#5833C2
hypercenter#536D3
hypocenterC3#3201
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
3. #5234C2semidirect
4. #5228C2semidirect
5. #5227C2semidirect
6. #5052C2semidirect
7. #5051C2semidirect
8. #5049C2semidirect
9. #5048C2semidirect
10. #5047C2semidirect
11. #5047C2semidirect
12. #5044C2semidirect
13. #5043C2semidirect
14. #5003C2semidirect
15. #4996C2semidirect
61. #536D3semidirect
373. C3#3201semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58332
#52342
#52282
#52272
#50522
#50512
#50492
#50482
#50472
#50442
#50432
#50032
#49962
#32013