Finite Group Databasebeta

← #80032

Subgroups — #80032

Order 384.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5812C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic
4#5228C211yesmaximal, characteristic, complemented
5#5166C211yesmaximal, characteristic, complemented
6#5166C211yesmaximal, characteristic, complemented
7#5164C211yesmaximal, characteristic, complemented
8Dic3×SD16C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic
10#4793C211yesmaximal, characteristic, complemented
11#4791C211yesmaximal, characteristic, complemented
12#4790C211yesmaximal, characteristic, complemented
13#4789C211yesmaximal, characteristic, complemented
14#4723C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3121—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×#121C2211yescharacteristic
21Q8×Dic3—21no—
22Q8×Dic3—21no—
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#216C2211yescharacteristic
29#936—21no—
30#936C2211yescharacteristic
31#935—21no—
32D4×Dic3C2211yescharacteristic
33#926—21no—
34#926—21no—
35#926C2211yescharacteristic
36#926C2211yescharacteristic
37#924C2211yescharacteristic
38#888C2211yescharacteristic
39#887C2211yescharacteristic
40#879—21no—
41#878—21no—
42C3×#103C2211yescharacteristic
43C3×#103C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#830C2211yescharacteristic
50#818C2211yescharacteristic
51#817C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#542—31no—
61#519—31no—
62#485—31no—
63#482—31no—
64#477—31no—
65#475—31no—
66#464—31no—
67#464—31no—
68#451—31no—
69#448D311yescharacteristic
70#441—31no—
71#440—31no—
72C4×Q16—31no—
73C4×SD16—31no—
74#433—31no—
75C2×C3×Q8—22no—
76C2×C3×Q8—22no—
77C2×C3×Q8—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C22×Dic3—21no—
85C2×Dic6—21no—
86C2×Dic6C2311yescharacteristic
87C3×SD16—22no—
88C3×SD16—22no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93#217—21no—
94#217—21no—
95#217—21no—
96#217—21no—
97#217—21no—
98#217D411yescharacteristic
99#217D411yescharacteristic
100#216—22no—
101#216—22no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3D411yescharacteristic
107C4⋊Dic3D411yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C22×Q8C2×D311yescharacteristic
125C2×Q16—31no—
126C2×Q16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×SD16C2×D311yescharacteristic
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×Q8—61no—
143C4×Q8—61no—
144C4×D4—31no—
145#117—31no—
146#117—31no—
147#117—61no—
148#117—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#103C2×D311yescharacteristic
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—24no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8#4112yes—
173C3×Q8#4112yes—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—81no—
177C2×C3×C4—24no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—81no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194Dic6—21no—
195Dic6—21no—
196Dic6—81no—
197C3×C8—21no—
198C3×C8—21no—
199C3⋊C8—21no—
200C3⋊C8—21no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—31no—
205C2×Q8—61no—
206C2×Q8C22×D311yescharacteristic
207C2×Q8C22×D311yescharacteristic
208C2×D4—21no—
209C2×D4C22×D311yescharacteristic
210C22×C4—21no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213Q16—62no—
214Q16—62no—
215SD16—22no—
216SD16—22no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8C22×D311yescharacteristic
220C2×C8C22×D311yescharacteristic
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⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4C22×D311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—31no—
236C22⋊C4—31no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C42—31no—
243C42—31no—
244C42—61no—
245C42—61no—
246C42—61no—
247C42—61no—
248C42—61no—
249C42—61no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4C2×#4111yescharacteristic
262C3×C4C2×#4111yescharacteristic
263Dic3—21no—
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—41no—
270Dic3—41no—
271C23—21no—
272C23D3×D411yescharacteristic
273Q8—24no—
274Q8—21no—
275Q8—21no—
276Q8—61no—
277Q8—61no—
278Q8—241no—
279Q8#23712yes—
280Q8#23712yes—
281D4—21no—
282D4—21no—
283D4—81no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—241no—
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—41no—
306C2×C3—81no—
307C2×C3#52511yescharacteristic
308C2×C3#57711yescharacteristic
309C2×C3#57811yescharacteristic
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22C2×D3×D411yescharacteristic
317C4—22no—
318C4—22no—
319C4—21no—
320C4—41no—
321C4—61no—
322C4—61no—
323C4—61no—
324C4—61no—
325C4—121no—
326C4—121no—
327C4—121no—
328C4—121no—
329C4C2×#23711yescharacteristic
330C4C2×#23711yescharacteristic
331C3#312111yescharacteristic, complemented, Sylow
332C2—41no—
333C2—81no—
334C2#558611yescharacteristic
335C2#576211yescharacteristic
336C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radicalC3×#121C22
hypercenter#448D3
hypocenterC3#3121
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #5812C2semidirect
4. #5228C2semidirect
5. #5166C2semidirect
6. #5166C2semidirect
7. #5164C2semidirect
8. Dic3×SD16C2semidirect
10. #4793C2semidirect
11. #4791C2semidirect
12. #4790C2semidirect
13. #4789C2semidirect
14. #4723C2semidirect
15. #4720C2semidirect
331. C3#3121semidirect

Maximal subgroups

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

subgroupindex
#58122
#56032
C3×#4482
#52282
#51662
#51642
Dic3×SD162
#50352
#47932
#47912
#47902
#47892
#47232
#47202
#31213