Finite Group Databasebeta

← #78907

Subgroups — #78907

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5824C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic, complemented
3C3×#417C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5137C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4971C211yesmaximal, characteristic, complemented
8#4816C211yesmaximal, characteristic, complemented
9#4805C211yesmaximal, characteristic, complemented
10#4795C211yesmaximal, characteristic, complemented
11#4792C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4771C211yesmaximal, characteristic, complemented
14#4761C211yesmaximal, characteristic, complemented
15#4758C211yesmaximal, characteristic, complemented
16#3331—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21#946C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#216C2211yescharacteristic
24#939—21no—
25#935—21no—
26D4×Dic3C2211yescharacteristic
27C2×#214C2211yescharacteristic
28#929—21no—
29C4×#67—21no—
30C4×#67C2211yescharacteristic
31#924—21no—
32#924C2211yescharacteristic
33#924C2211yescharacteristic
34C2×Dic3⋊C4—21no—
35C2×Dic12C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#897C2211yescharacteristic
38#896—21no—
39D3×C4⋊C4C2211yescharacteristic
40#889—21no—
41#888C2211yescharacteristic
42#887C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#808C2211yescharacteristic
60#554—31no—
61#547—31no—
62#487—31no—
63#483—31no—
64#483—31no—
65#476—31no—
66#470—31no—
67#470—31no—
68#462—31no—
69#452—31no—
70#452—31no—
71C4×SD16—31no—
72#436—31no—
73#417—31no—
74#417D311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86C22×Dic3—21no—
87#237—22no—
88#237—41no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×M16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100#217—21no—
101#217—21no—
102#217—21no—
103#217D411yescharacteristic
104#216—41no—
105#214—22no—
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⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125Dic12—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136#125—61no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4C2×D311yescharacteristic
153#106—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#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×C4—81no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—22no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—81no—
201Dic6—21no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—81no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—22no—
211#41—41no—
212#41—62no—
213#41—121no—
214C2×Q8—31no—
215C2×Q8—31no—
216C2×Q8—61no—
217C2×Q8C22×D311yescharacteristic
218C2×D4—21no—
219C2×D4—31no—
220C2×D4—61no—
221C2×D4C22×D311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—121no—
230Q16—121no—
231SD16—62no—
232M16—41no—
233M16—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4C22×D311yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—31no—
254C22⋊C4—31no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C42—31no—
261C42—61no—
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C22×D411yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—21no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4C2×#4111yescharacteristic
277C3×C4H(5,𝔽2)11yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3#13712yes—
285C23—21no—
286C23—31no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—61no—
291Q8—61no—
292Q8—61no—
293Q8—61no—
294Q8—241no—
295D4—21no—
296D4—42no—
297D4—62no—
298D4—122no—
299D4#23512yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—42no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—81no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—241no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—41no—
326C2×C3—42no—
327C2×C3#54611yescharacteristic
328C2×C3C2×#13711yescharacteristic
329C2×C3#57711yescharacteristic
330D3—41no—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—32no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—61no—
347C4—61no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23511yescharacteristic
353C4#100411yescharacteristic
354C3#333111yescharacteristic, complemented, Sylow
355C2—41no—
356C2—42no—
357C2—121no—
358C2#549011yescharacteristic
359C2#577511yescharacteristic
360C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
prime radical#5824C2
hypercenter#417D3
hypocenterC3#3331
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #5511C2semidirect
3. C3×#417C2semidirect
4. #5239C2semidirect
5. #5137C2semidirect
6. #5128C2semidirect
7. #4971C2semidirect
8. #4816C2semidirect
9. #4805C2semidirect
10. #4795C2semidirect
11. #4792C2semidirect
12. #4783C2semidirect
13. #4771C2semidirect
14. #4761C2semidirect
15. #4758C2semidirect
74. #417D3semidirect
354. C3#3331semidirect

Maximal subgroups

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

subgroupindex
#58242
#55112
C3×#4172
#52392
#51372
#51282
#49712
#48162
#48052
#47952
#47922
#47832
#47712
#47612
#47582
#33313