Finite Group Databasebeta

← #85878

Subgroups — #85878

Order 384.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#519C211yesmaximal, characteristic, complemented
2#5817C211yesmaximal, characteristic, complemented
3Q8×#67C211yesmaximal, characteristic, complemented
4#5584C213yesmaximal, complemented
5#5582C213yesmaximal, complemented
6#5578C213yesmaximal, complemented
7#5576C211yesmaximal, characteristic, complemented
8C4×#239C211yesmaximal, characteristic, complemented
9#5572C211yesmaximal, characteristic, complemented
10#5568C213yesmaximal, complemented
11Q8×Dic6C211yesmaximal, characteristic, complemented
12#5534C213yesmaximal, complemented
13#5533C213yesmaximal, complemented
14#5532C213yesmaximal, complemented
15#5528C213yesmaximal, complemented
16#3518—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×Q8C2213yes—
22C3×C4×Q8C2211yescharacteristic
23C3×C4×Q8C2213yes—
24C3×#117C2213yes—
25C3×#117C2213yes—
26#947C2213yes—
27#946C2213yes—
28#946C2213yes—
29Q8×Dic3C2211yescharacteristic
30Q8×Dic3C2211yescharacteristic
31#944C2213yes—
32#930C2213yes—
33C4×#67C2213yes—
34C4×#67C2213yes—
35#924C2213yes—
36#898C2213yes—
37#898C2216yes—
38#897C2213yes—
39#896C2213yes—
40#896C2216yes—
41#895C2213yes—
42#894C2213yes—
43#893C2213yes—
44#892C2213yes—
45#892C2213yes—
46#889C2213yes—
47#888C2213yes—
48#888C2213yes—
49#887C2213yes—
50#885C2211yescharacteristic
51#885C2216yes—
52#884C2213yes—
53#883C2213yes—
54#879C2213yes—
55#876C2216yes—
56#875C2216yes—
57#875C2213yes—
58C4×D12C2213yes—
59C4×D12C2213yes—
60#872C2213yes—
61#872C2216yes—
62C42×D3C2213yes—
63#869C2213yes—
64C4×Dic6C2213yes—
65C4×Dic6C2216yes—
66C4×Dic6C2213yes—
67Q82—31no—
68#556—33no—
69#556—33no—
70#556—33no—
71#555—33no—
72D4×Q8—31no—
73#548—31no—
74#548—31no—
75#542—33no—
76#542—33no—
77#533—33no—
78#529—33no—
79#519—31no—
80#519D311yescharacteristic, complemented
81C4×#41—31no—
82C2×C3×Q8—24no—
83C2×C3×Q8C2313yes—
84C2×C3×Q8C2311yescharacteristic
85C22×C3×C4C2313yes—
86C2×#67C2311yescharacteristic
87C2×#67C2313yes—
88#239—24no—
89D3×Q8—24no—
90C2×D12C2313yes—
91C2×D12C2313yes—
92C2×C4×D3C2313yes—
93C2×C4×D3C2313yes—
94C2×C4×D3C2313yes—
95C2×C4×D3C2313yes—
96C2×Dic6C2311yescharacteristic
97C2×Dic6C2313yes—
98C2×Dic6C2316yes—
99C3×C4⋊C4C2313yes—
100C3×C4⋊C4C2313yes—
101C3×C4⋊C4C2313yes—
102C3×C4⋊C4C2313yes—
103C3×C22⋊C4C2313yes—
104C3×C22⋊C4C2311yescharacteristic
105C3×C42C2313yes—
106C3×C42C2313yes—
107C3×C42C2316yes—
108#217C2311yescharacteristic
109#217C2313yes—
110(C2×D3)⋊C4C2311yescharacteristic
111(C2×D3)⋊C4C2313yes—
112(C2×D3)⋊C4C2316yes—
113(C2×D3)⋊C4C2316yes—
114(C2×D3)⋊C4C2311yescharacteristic
115(C2×D3)⋊C4C2313yes—
116(C2×D3)⋊C4C2316yes—
117(C2×D3)⋊C4C2313yes—
118(C2×D3)⋊C4C2313yes—
119C4⋊Dic3C2313yes—
120C4⋊Dic3C2313yes—
121C4⋊Dic3C2313yes—
122C4⋊Dic3C2313yes—
123Dic3⋊C4C2313yes—
124Dic3⋊C4C2313yes—
125Dic3⋊C4C2313yes—
126Dic3⋊C4C2313yes—
127Dic3⋊C4C2316yes—
128Dic3⋊C4C2313yes—
129Dic3⋊C4C2313yes—
130C4×Dic3C2313yes—
131C4×Dic3C2316yes—
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2311yescharacteristic
135C2×#41—31no—
136C22×Q8—31no—
137C22×Q8C2×D311yescharacteristic
138C4⋊Q8—33no—
139C4⋊Q8—33no—
140C4⋊Q8—33no—
141C4⋊Q8—33no—
142#126—33no—
143#126—36no—
144#126—33no—
145#126—36no—
146#125—33no—
147#124—33no—
148#124—31no—
149#124—36no—
150#124—33no—
151#124—36no—
152#123—33no—
153#123—33no—
154C22⋊Q8—33no—
155C22⋊Q8—33no—
156C22⋊Q8—33no—
157C22⋊Q8—33no—
158C22⋊Q8—36no—
159C22⋊Q8—33no—
160#121—33no—
161#121—33no—
162#121—33no—
163C4×Q8—31no—
164C4×Q8—31no—
165C4×Q8—33no—
166C4×Q8—33no—
167C4×Q8—33no—
168C4×Q8—36no—
169C4×Q8C2×D311yescharacteristic
170C4×Q8C2×D313yes—
171C4×Q8C2×D311yescharacteristic
172C4×Q8C2×D313yes—
173C4×D4—33no—
174C4×D4—33no—
175C4×D4—33no—
176C4×D4—33no—
177C4×D4—33no—
178#117—33no—
179#117—33no—
180#117—33no—
181#117—36no—
182#117C2×D313yes—
183#117C2×D313yes—
184C2×C42—33no—
185C23×C3C2411yescharacteristic
186C22×D3C2411yescharacteristic
187C22×D3C2413yes—
188C3×Q8—26no—
189C3×Q8#4114yes—
190C2×C3×C4—26no—
191C2×C3×C4C2413yes—
192C2×C3×C4C2411yescharacteristic
193C2×C3×C4C2413yes—
194C2×C3×C4C2413yes—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2413yes—
197#67—22no—
198#67—43no—
199C2×Dic3C2411yescharacteristic
200C2×Dic3C2411yescharacteristic
201C2×Dic3C2413yes—
202C2×Dic3C2413yes—
203C2×Dic3C2413yes—
204C2×Dic3C2411yescharacteristic
205D12—26no—
206D12—43no—
207C4×D3—26no—
208C4×D3—26no—
209C4×D3—26no—
210C4×D3—43no—
211Dic6—26no—
212Dic6—22no—
213Dic6—46no—
214#41—64no—
215C2×Q8—24no—
216C2×Q8—31no—
217C2×Q8—33no—
218C2×Q8—36no—
219C2×Q8—64no—
220C2×Q8C22×D313yes—
221C2×Q8C22×D311yescharacteristic
222C2×D4—31no—
223C2×D4—33no—
224C2×D4—33no—
225C2×D4—33no—
226C22×C4—33no—
227C22×C4—33no—
228C22×C4—33no—
229C22×C4—33no—
230C22×C4C22×D313yes—
231C4⋊C4—33no—
232C4⋊C4—33no—
233C4⋊C4—33no—
234C4⋊C4—33no—
235C4⋊C4—33no—
236C4⋊C4—33no—
237C4⋊C4—33no—
238C4⋊C4—33no—
239C4⋊C4—36no—
240C4⋊C4—33no—
241C4⋊C4—33no—
242C4⋊C4C22×D313yes—
243C4⋊C4C22×D313yes—
244C4⋊C4C22×D313yes—
245C4⋊C4C22×D313yes—
246C22⋊C4—31no—
247C22⋊C4—33no—
248C22⋊C4—33no—
249C22⋊C4—31no—
250C22⋊C4—36no—
251C22⋊C4—36no—
252C22⋊C4—36no—
253C22⋊C4—31no—
254C22⋊C4—33no—
255C22⋊C4—33no—
256C22⋊C4—33no—
257C22⋊C4C22×D313yes—
258C22⋊C4C22×D311yescharacteristic
259C42—31no—
260C42—33no—
261C42—33no—
262C42—36no—
263C42—33no—
264C42C22×D313yes—
265C42C22×D316yes—
266C42C22×D313yes—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3C2511yescharacteristic
271C2×D3—21no—
272C2×D3—23no—
273C2×D3—23no—
274C2×D3—21no—
275C2×D3—23no—
276C2×D3—21no—
277C3×C4—23no—
278C3×C4—21no—
279C3×C4—23no—
280C3×C4—23no—
281C3×C4—21no—
282C3×C4C2×#4116yes—
283Dic3—23no—
284Dic3—21no—
285Dic3—21no—
286Dic3—23no—
287Dic3—23no—
288Dic32−1+412yes—
289C23—31no—
290C23—33no—
291C23C23×D311yescharacteristic
292Q8—26no—
293Q8—62no—
294Q8—66no—
295Q8—126no—
296Q8#23514yes—
297D4—62no—
298D4—66no—
299D4—123no—
300D4—123no—
301C2×C4—26no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—33no—
306C2×C4—33no—
307C2×C4—33no—
308C2×C4—66no—
309C2×C4—66no—
310C2×C4—66no—
311C2×C4—123no—
312C2×C4C23×D313yes—
313C2×C4C23×D311yescharacteristic
314C2×C4C23×D311yescharacteristic
315C2×C4C23×D313yes—
316C2×C4C23×D313yes—
317C2×C4C23×D313yes—
318C2×C3—41no—
319C2×C3C22×#4111yescharacteristic
320C2×C3C2×2−1+411yescharacteristic
321C2×C3#58511yescharacteristic
322D3—41no—
323D3—43no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—61no—
328C22—63no—
329C22—63no—
330C22—61no—
331C22—63no—
332C22—61no—
333C22C24×D311yescharacteristic
334C4—23no—
335C4—23no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—32no—
340C4—61no—
341C4—63no—
342C4—61no—
343C4—63no—
344C4—63no—
345C4C2×#23516yes—
346C3#351811yescharacteristic, complemented, Sylow
347C2—41no—
348C2—121no—
349C2—123no—
350C2C22×#23511yescharacteristic
351C2D3×2−1+411yescharacteristic
352C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#519C2
hypercenter#519D3
hypocenterC3#3518
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C3×#519C2semidirect
2. #5817C2semidirect
3. Q8×#67C2semidirect
4. #5584C2semidirect
5. #5582C2semidirect
6. #5578C2semidirect
7. #5576C2semidirect
8. C4×#239C2semidirect
9. #5572C2semidirect
10. #5568C2semidirect
11. Q8×Dic6C2semidirect
12. #5534C2semidirect
13. #5533C2semidirect
14. #5532C2semidirect
15. #5528C2semidirect
80. #519D3semidirect
346. C3#3518semidirect

Maximal subgroups

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

subgroupindex
C3×#5192
#58172
Q8×#672
#55842
#55822
#55782
#55762
C4×#2392
#55722
#55682
Q8×Dic62
#55342
#55332
#55322
#55282
#35183