Finite Group Databasebeta

← #83621

Subgroups — #83621

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#553C211yesmaximal, characteristic, complemented
2#5552C211yesmaximal, characteristic, complemented
3#5068C211yesmaximal, characteristic, complemented
4#5063C211yesmaximal, characteristic, complemented
5#5042C211yesmaximal, characteristic, complemented
6#5040C211yesmaximal, characteristic, complemented
7#5039C211yesmaximal, characteristic, complemented
8#5037C211yesmaximal, characteristic, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10#5016C211yesmaximal, characteristic, complemented
11#5012C211yesmaximal, characteristic, complemented
12#5004C211yesmaximal, characteristic, complemented
13#5003C211yesmaximal, characteristic, complemented
14#4971C211yesmaximal, characteristic, complemented
15#4966C211yesmaximal, characteristic, complemented
16#3326—31nomaximal, Sylow
17C2×#235C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C3×#126—21no—
20C3×#125C2211yescharacteristic
21C3×#123—21no—
22C3×#121—21no—
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×C4×D4—21no—
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C3×#117C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30#939—21no—
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic
33C2×#213C2211yescharacteristic
34#930C2211yescharacteristic
35#924C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C2×#208C2211yescharacteristic
38#896—21no—
39#885—21no—
40#67⋊C4—21no—
41#875C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43#832C2211yescharacteristic
44#830C2211yescharacteristic
45#830C2211yescharacteristic
46#810C2211yescharacteristic
47#810C2211yescharacteristic
48#810C2211yescharacteristic
49#810C2211yescharacteristic
50#810C2211yescharacteristic
51#810C2211yescharacteristic
52#809C2211yescharacteristic
53#809C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59C4×C3⋊C8C2211yescharacteristic
60#553D311yescharacteristic, complemented
61#548—31no—
62#488—31no—
63#483—31no—
64#482—31no—
65#479—31no—
66#469—31no—
67#468—31no—
68#463—31no—
69#449—31no—
70#448—31no—
71C4×SD16—31no—
72#436—31no—
73#419—31no—
74#417—31no—
75C2×C3×D4—21no—
76C2×C3×D4—21no—
77C2×C3×D4D411yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#235—22no—
87#235—41no—
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×Dic6—22no—
91C2×Dic6—21no—
92C2×Dic6—41no—
93C2×Dic6C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4—21no—
98C3×C4⋊C4D411yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106C3×C22⋊C4D411yescharacteristic
107C3×C22⋊C4D411yescharacteristic
108C3×C42—21no—
109C3×C42C2311yescharacteristic
110#217—21no—
111#214—22no—
112#214—41no—
113#213—41no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121#208—41no—
122#208—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41—31no—
128C22×Q8—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×D8—31no—
132C2×M16—31no—
133C2×M16—31no—
134#126—21no—
135#125C2×D311yescharacteristic
136#124—31no—
137#124—61no—
138#123—21no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141#121—21no—
142#121—31no—
143#121—61no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×D4—21no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150C4×D4C2×D311yescharacteristic
151#117C2×D311yescharacteristic
152C2×C4⋊C4C2×D311yescharacteristic
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—41no—
177C3×D4—42no—
178C3×D4—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199D12—21no—
200D12—21no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—42no—
206Dic6#4112yes—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—62no—
212#41—121no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—121no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4#6711yescharacteristic
224C2×D4C22×D311yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231SD16—62no—
232SD16—121no—
233D8—121no—
234M16—121no—
235M16—121no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—31no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4#6711yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C4⋊C4C22×D311yescharacteristic
249C4⋊C4C22×D311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4#6711yescharacteristic
259C22⋊C4#6711yescharacteristic
260C42—21no—
261C42—61no—
262C42C22×D311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3C22×D411yescharacteristic
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4C2×#4111yescharacteristic
284C3×C4H(5,𝔽2)11yescharacteristic
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288C23—21no—
289C23—61no—
290C23C2×#6711yescharacteristic
291C23C2×#6711yescharacteristic
292Q8—32no—
293Q8—61no—
294Q8—61no—
295Q8—61no—
296Q8—122no—
297D4—21no—
298D4—21no—
299D4—41no—
300D4—42no—
301D4—61no—
302D4—61no—
303D4—81no—
304D4—122no—
305D4—122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4C2×#6711yescharacteristic
321C2×C4C2×#6711yescharacteristic
322C2×C4C2×#6711yescharacteristic
323C2×C4C2×#6711yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—61no—
326C8—61no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3#54611yescharacteristic
333C2×C3#57511yescharacteristic
334C2×C3#57711yescharacteristic
335D3—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C22×#6711yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—62no—
357C4—121no—
358C4—121no—
359C4D3×#4111yescharacteristic
360C4#100911yescharacteristic
361C3#332611yescharacteristic, complemented, Sylow
362C2—41no—
363C2—41no—
364C2—81no—
365C2—241no—
366C2#582211yescharacteristic
367C2#582911yescharacteristic
368C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#553C2
socleC22×C3C22×D4
hypercenter#553D3
hypocenterC3#3326
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. C3×#553C2semidirect
2. #5552C2semidirect
3. #5068C2semidirect
4. #5063C2semidirect
5. #5042C2semidirect
6. #5040C2semidirect
7. #5039C2semidirect
8. #5037C2semidirect
9. C4×#214C2semidirect
10. #5016C2semidirect
11. #5012C2semidirect
12. #5004C2semidirect
13. #5003C2semidirect
14. #4971C2semidirect
15. #4966C2semidirect
60. #553D3semidirect
361. C3#3326semidirect

Maximal subgroups

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

subgroupindex
C3×#5532
#55522
#50682
#50632
#50422
#50402
#50392
#50372
C4×#2142
#50162
#50122
#50042
#50032
#49712
#49662
#33263