Finite Group Databasebeta

← #79473

Subgroups — #79473

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5565C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5121C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7#5106C211yesmaximal, characteristic, complemented
8#4847C211yesmaximal, characteristic, complemented
9#4846C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4737C211yesmaximal, characteristic, complemented
12#4737C211yesmaximal, characteristic, complemented
13#4736C211yesmaximal, characteristic, complemented
14#4711C211yesmaximal, characteristic, complemented
15#4702C211yesmaximal, characteristic, complemented
16#3347—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C22×C3×C8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#939—21no—
21#938—21no—
22#930C2211yescharacteristic
23#930C2211yescharacteristic
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#926C2211yescharacteristic
27#926C2211yescharacteristic
28#926C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×D24C2211yescharacteristic
31#898—21no—
32#892—21no—
33#885—21no—
34#885—21no—
35#883—21no—
36#67⋊C4—21no—
37C4×D12C2211yescharacteristic
38C4×D12C2211yescharacteristic
39#870C2211yescharacteristic
40#869C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×C22⋊C8C2211yescharacteristic
43C3×C22⋊C8C2211yescharacteristic
44C3×#97C2211yescharacteristic
45#821C2211yescharacteristic
46#821C2211yescharacteristic
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#818C2211yescharacteristic
53#818C2211yescharacteristic
54#818C2211yescharacteristic
55#818C2211yescharacteristic
56#817C2211yescharacteristic
57#817C2211yescharacteristic
58#817C2211yescharacteristic
59#817C2211yescharacteristic
60#555—31no—
61#553—31no—
62#501—31no—
63#482—31no—
64#482—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#474—31no—
69#469—31no—
70#466—31no—
71#442—31no—
72#435D311yescharacteristic, complemented
73#428—31no—
74#427—31no—
75C2×C3×D4D411yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×Dic6—21no—
88C3×M16—22no—
89C2×C3×C8—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4D411yescharacteristic
95C3×C22⋊C4D411yescharacteristic
96C3×C22⋊C4D411yescharacteristic
97C3×C42C2311yescharacteristic
98#217—21no—
99#217—21no—
100#217—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120D24—22no—
121C2×D8—31no—
122C2×M16C2×D311yescharacteristic
123C22×C8C2×D311yescharacteristic
124C4⋊Q8—31no—
125#126—61no—
126#125—31no—
127#124—61no—
128#124—61no—
129#123—61no—
130#121—31no—
131#121—31no—
132#121—31no—
133#121—31no—
134#121—61no—
135#121—61no—
136C4×D4—31no—
137C4×D4—31no—
138C4×D4—61no—
139C4×D4C2×D311yescharacteristic
140#117—31no—
141#117—31no—
142#117—31no—
143#117—61no—
144C2×C4⋊C4—31no—
145#107—31no—
146#107—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#106—31no—
151#106—31no—
152#106—31no—
153#106—31no—
154C4⋊C8C2×D311yescharacteristic
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161C22⋊C8C2×D311yescharacteristic
162C22⋊C8C2×D311yescharacteristic
163#97C2×D311yescharacteristic
164C23×C3C2×D411yescharacteristic
165C23×C3C2×D411yescharacteristic
166C22×D3—21no—
167C22×D3—21no—
168C3×D4—41no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4C2×D411yescharacteristic
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2×D411yescharacteristic
177C2×C3×C4C2411yescharacteristic
178#67—42no—
179#67—81no—
180#67—81no—
181#67—81no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—81no—
189D12—21no—
190D12—21no—
191D12—21no—
192D12—21no—
193C4×D3—42no—
194C4×D3—81no—
195Dic6—81no—
196C3×C8—21no—
197C3×C8—21no—
198C3×C8—21no—
199C3×C8#4112yes—
200C2×Q8—61no—
201C2×D4—31no—
202C2×D4—31no—
203C2×D4—61no—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4D1211yescharacteristic
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D311yescharacteristic
213D8—62no—
214M16—22no—
215C2×C8—22no—
216C2×C8C22×D311yescharacteristic
217C2×C8C22×D311yescharacteristic
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4D1211yescharacteristic
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4D1211yescharacteristic
241C22⋊C4D1211yescharacteristic
242C42—61no—
243C42—61no—
244C42—61no—
245C42C22×D311yescharacteristic
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3C22×D411yescharacteristic
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4C2×#4111yescharacteristic
264C3×C42−1+411yescharacteristic
265Dic3—41no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—61no—
272C23—61no—
273C23C2×D1211yescharacteristic
274C23C2×D1211yescharacteristic
275Q8—241no—
276D4—41no—
277D4—61no—
278D4—61no—
279D4—61no—
280D4—61no—
281D4—122no—
282D4—241no—
283D4—241no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—122no—
296C2×C4—241no—
297C2×C4—241no—
298C2×C4C2×D1211yescharacteristic
299C2×C4C2×D1211yescharacteristic
300C2×C4C2×D1211yescharacteristic
301C2×C4C2×D1211yescharacteristic
302C2×C4C23×D311yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8#23712yes—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3#54711yescharacteristic
310C2×C3#57511yescharacteristic
311C2×C3#57611yescharacteristic
312D3—81no—
313D3—81no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C22×D1211yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—121no—
332C4—121no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4—121no—
337C4C2×#23711yescharacteristic
338C4#101011yescharacteristic
339C3#334711yescharacteristic, complemented, Sylow
340C2—41no—
341C2—41no—
342C2—241no—
343C2—241no—
344C2#555511yescharacteristic
345C2#574811yescharacteristic
346C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3347
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #5565C2semidirect
2. #5563C2semidirect
3. C3×#435C2semidirect
4. #5135C2semidirect
5. #5121C2semidirect
6. #5115C2semidirect
7. #5106C2semidirect
8. #4847C2semidirect
9. #4846C2semidirect
10. #4737C2semidirect
11. #4737C2semidirect
12. #4737C2semidirect
13. #4736C2semidirect
14. #4711C2semidirect
15. #4702C2semidirect
72. #435D3semidirect
339. C3#3347semidirect

Maximal subgroups

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

subgroupindex
#55652
#55632
C3×#4352
#51352
#51212
#51152
#51062
#48472
#48462
#47372
#47362
#47112
#47022
#33473