Finite Group Databasebeta

← #80271

Subgroups — #80271

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5825C211yesmaximal, characteristic
2#5625C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic
4#5239C211yesmaximal, characteristic
5#5183C211yesmaximal, characteristic, complemented
6#5182C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5166C211yesmaximal, characteristic, complemented
9#5050C211yesmaximal, characteristic, complemented
10#4797C211yesmaximal, characteristic, complemented
11#4795C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4767C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3223—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23#944C2211yescharacteristic
24#944C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#216C2211yescharacteristic
29#936C2211yescharacteristic
30#936C2211yescharacteristic
31#935—21no—
32#935—21no—
33C2×#214C2211yescharacteristic
34#926C2211yescharacteristic
35#924—21no—
36#924—21no—
37#924C2211yescharacteristic
38C2×Dic12C2211yescharacteristic
39C2×Dic12C2211yescharacteristic
40#890—21no—
41#890C2211yescharacteristic
42#889—21no—
43#887C2211yescharacteristic
44#879—21no—
45#879—21no—
46C3×#103C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#810C2211yescharacteristic
59#810C2211yescharacteristic
60#541—31no—
61#536—31no—
62#497—31no—
63#494—31no—
64#491—31no—
65#488—31no—
66#464—31no—
67#464—31no—
68#462—31no—
69#462—31no—
70#452—31no—
71#452D311yescharacteristic
72#451—31no—
73#451—31no—
74#433—31no—
75C3×#41—41no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C22×Dic3—21no—
84C2×Dic6—21no—
85C2×Dic6—41no—
86C2×Dic6—41no—
87C2×Dic6C2311yescharacteristic
88C2×Dic6C2311yescharacteristic
89C2×Dic6C2311yescharacteristic
90C3×Q16—41no—
91C3×SD16—41no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217D411yescharacteristic
103#217D411yescharacteristic
104#216—41no—
105#216—41no—
106#216—41no—
107#214—41no—
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3D411yescharacteristic
111C4⋊Dic3D411yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129Dic12—41no—
130C2×#41C2×D311yescharacteristic
131C22×Q8—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×Q16—31no—
136C2×Q16—31no—
137C2×Q16C2×D311yescharacteristic
138C2×SD16—31no—
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#126—61no—
144#126—61no—
145#125—31no—
146#125—61no—
147#125—61no—
148#124—31no—
149#124—31no—
150#123—61no—
151#123—61no—
152C22⋊Q8—31no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C22⋊Q8C2×D311yescharacteristic
156C4×Q8—31no—
157C4×Q8—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—42no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—42no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—21no—
204Dic6—21no—
205Dic6—42no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—41no—
211#41—41no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×Q8—61no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×Q8C22×D311yescharacteristic
220C2×D4—21no—
221C2×D4C22×D311yescharacteristic
222C22×C4—21no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225Q16—41no—
226Q16—121no—
227Q16—121no—
228Q16—121no—
229Q16—121no—
230Q16—121no—
231SD16—41no—
232SD16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8C22×D311yescharacteristic
236C2×C8C22×D311yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
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⋊C4—61no—
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—31no—
255C22⋊C4—31no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C42—31no—
262C42—31no—
263C42—61no—
264C42—61no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C22×D411yescharacteristic
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4H(5,𝔽2)11yescharacteristic
277C3×C4H(5,𝔽2)11yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286C23—21no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—61no—
293Q8—61no—
294Q8—61no—
295Q8—61no—
296Q8—61no—
297Q8—61no—
298Q8—122no—
299D4—21no—
300D4—21no—
301D4—42no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—42no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—122no—
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—41no—
324C2×C3—81no—
325C2×C3#53511yescharacteristic
326C2×C3#57811yescharacteristic
327C2×C3#57811yescharacteristic
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C2×D3×D411yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—61no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4#100411yescharacteristic
348C4#100411yescharacteristic
349C3#322311yescharacteristic, complemented, Sylow
350C2—41no—
351C2—81no—
352C2#559211yescharacteristic
353C2#577111yescharacteristic
354C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#452D3
hypocenterC3#3223
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #5625C2semidirect
5. #5183C2semidirect
6. #5182C2semidirect
7. #5167C2semidirect
8. #5166C2semidirect
9. #5050C2semidirect
10. #4797C2semidirect
11. #4795C2semidirect
12. #4768C2semidirect
13. #4767C2semidirect
14. #4739C2semidirect
15. #4720C2semidirect
349. C3#3223semidirect

Maximal subgroups

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

subgroupindex
#58252
#56252
C3×#4522
#52392
#51832
#51822
#51672
#51662
#50502
#47972
#47952
#47682
#47672
#47392
#47202
#32233