Finite Group Databasebeta

← #78849

Subgroups — #78849

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5824C211yesmaximal, characteristic
2#5533C211yesmaximal, characteristic, complemented
3C3×#416C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5116C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7#4996C211yesmaximal, characteristic, complemented
8#4814C211yesmaximal, characteristic, complemented
9#4813C211yesmaximal, characteristic, complemented
10#4795C211yesmaximal, characteristic, complemented
11#4792C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4772C211yesmaximal, characteristic, complemented
14#4759C211yesmaximal, characteristic, complemented
15#4758C211yesmaximal, characteristic, complemented
16#3302—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#946C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24#939—21no—
25#935—21no—
26D4×Dic3C2211yescharacteristic
27C2×#214C2211yescharacteristic
28#929—21no—
29C4×#67C2211yescharacteristic
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C2×Dic3⋊C4—21no—
33C2×Dic12C2211yescharacteristic
34C2×D3×C8C2211yescharacteristic
35#897C2211yescharacteristic
36#892C2211yescharacteristic
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#885—21no—
40#879—21no—
41#876—21no—
42C4×Dic6—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#818C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#556—31no—
61#547—31no—
62#487—31no—
63#485—31no—
64#485—31no—
65#474—31no—
66#467—31no—
67#467—31no—
68#462—31no—
69#452—31no—
70#452—31no—
71C4×SD16—31no—
72C8×D4—31no—
73#416—31no—
74#416D311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67D411yescharacteristic
84C22×Dic3—21no—
85C22×Dic3—21no—
86#237—22no—
87#237—41no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2311yescharacteristic
92C2×C3×C8—41no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C42—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#216—41no—
104#214—22no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109C4⋊Dic3—21no—
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126D3×C8—41no—
127C2×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C22×C8—31no—
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136#126—61no—
137#126—61no—
138#124—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—61no—
148C4×D4—31no—
149C4×D4—31no—
150#117—31no—
151#117C2×D311yescharacteristic
152C2×C4⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×D4—21no—
176C3×D4—42no—
177C3×D4#4112yes—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—42no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—22no—
187#67—42no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C4×D3—21no—
198C4×D3—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—22no—
209#41—41no—
210#41—62no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—61no—
215C2×Q8C22×D311yescharacteristic
216C2×D4—21no—
217C2×D4—31no—
218C2×D4—61no—
219C2×D4C22×D311yescharacteristic
220C22×C4—21no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—121no—
226Q16—121no—
227SD16—62no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8—41no—
231C2×C8—121no—
232C2×C8C22×D311yescharacteristic
233C2×C8C22×D311yescharacteristic
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—31no—
249C22⋊C4—31no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C42—21no—
257C42—31no—
258C42—61no—
259C42—61no—
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C2×#4111yescharacteristic
275C3×C4H(5,𝔽2)11yescharacteristic
276Dic3—21no—
277Dic3—21no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3#13512yes—
283C23—21no—
284C23—31no—
285C23D3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—122no—
293D4—21no—
294D4—42no—
295D4—62no—
296D4—122no—
297D4#23512yes—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—42no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D3×D411yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—21no—
318C8—21no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3—42no—
323C2×C3#54611yescharacteristic
324C2×C3C2×#13511yescharacteristic
325C2×C3#57811yescharacteristic
326D3—41no—
327C22—21no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22C2×D3×D411yescharacteristic
337C4—21no—
338C4—32no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23511yescharacteristic
349C4#100411yescharacteristic
350C3#330211yescharacteristic, complemented, Sylow
351C2—41no—
352C2—42no—
353C2—121no—
354C2#549011yescharacteristic
355C2D3×#13511yescharacteristic
356C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#416C2
socleC22×C3C22×D4
prime radical#5824C2
hypercenter#416D3
hypocenterC3#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #5533C2semidirect
3. C3×#416C2semidirect
4. #5239C2semidirect
5. #5116C2semidirect
6. C8×#67C2semidirect
7. #4996C2semidirect
8. #4814C2semidirect
9. #4813C2semidirect
10. #4795C2semidirect
11. #4792C2semidirect
12. #4783C2semidirect
13. #4772C2semidirect
14. #4759C2semidirect
15. #4758C2semidirect
74. #416D3semidirect
350. C3#3302semidirect

Maximal subgroups

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

subgroupindex
#58242
#55332
C3×#4162
#52392
#51162
C8×#672
#49962
#48142
#48132
#47952
#47922
#47832
#47722
#47592
#47582
#33023