Finite Group Databasebeta

← #83513

Subgroups — #83513

Order 384.

382 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
1C3×#547C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic, complemented
6#5232C211yesmaximal, characteristic, complemented
7#5090C211yesmaximal, characteristic, complemented
8#5082C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#5051C211yesmaximal, characteristic, complemented
11#5045C211yesmaximal, characteristic, complemented
12#5044C211yesmaximal, characteristic, complemented
13#5018C211yesmaximal, characteristic, complemented
14#5011C211yesmaximal, characteristic, complemented
15#5009C211yesmaximal, characteristic, complemented
16#3315—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×#123—21no—
21C3×#123—21no—
22C3×C22⋊Q8C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24C3×#121—21no—
25C3×C4×D4C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C2×C3×C4⋊C4—21no—
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30#935—21no—
31C2×#214C2211yescharacteristic
32#926C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34#924C2211yescharacteristic
35#924C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#889—21no—
39#879—21no—
40#877—21no—
41#870C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43#835C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic
54#810C2211yescharacteristic
55#808C2211yescharacteristic
56#807C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#552—31no—
61#547D311yescharacteristic
62#491—31no—
63#485—31no—
64#484—31no—
65#478—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#452—31no—
70#452—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C3×#41—22no—
76C3×#41—22no—
77C3×#41—41no—
78C3×#41—41no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4D411yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×Dic3—21no—
91C2×Dic6C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4D411yescharacteristic
102C3×C22⋊C4D411yescharacteristic
103C3×C42C2311yescharacteristic
104#217—21no—
105#217—21no—
106#217—21no—
107#216—41no—
108#216—41no—
109#214—22no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C2×#41C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8C2×D311yescharacteristic
135#126—61no—
136#125—31no—
137#125—61no—
138#123—21no—
139#123—21no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8C2×D311yescharacteristic
144C22⋊Q8C2×D311yescharacteristic
145#121—21no—
146C4×Q8—31no—
147C4×D4C2×D311yescharacteristic
148C4×D4C2×D311yescharacteristic
149#117—31no—
150#117—61no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—31no—
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#103—31no—
163#103—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—41no—
179C3×D4—42no—
180C3×D4—42no—
181C3×D4#4112yes—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2411yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—42no—
202Dic6—21no—
203Dic6—21no—
204C3⋊C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—22no—
209#41—22no—
210#41—41no—
211#41—41no—
212C2×Q8—31no—
213C2×Q8C22×D311yescharacteristic
214C2×Q8C22×D311yescharacteristic
215C2×D4—21no—
216C2×D4—21no—
217C2×D4#6711yescharacteristic
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—61no—
223C22×C4C22×D311yescharacteristic
224C22×C4C22×D311yescharacteristic
225Q16—121no—
226Q16—121no—
227SD16—62no—
228M16—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—121no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
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#6711yescharacteristic
248C4⋊C4C22×D311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4#6711yescharacteristic
256C22⋊C4#6711yescharacteristic
257C42—61no—
258C42—61no—
259C42C22×D311yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—21no—
284C23C2×#6711yescharacteristic
285C23C2×#6711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—61no—
291Q8—61no—
292D4—21no—
293D4—41no—
294D4—42no—
295D4—42no—
296D4#23712yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—122no—
312C2×C4C2×#6711yescharacteristic
313C2×C4C2×#6711yescharacteristic
314C2×C4C2×#6711yescharacteristic
315C2×C4C2×#6711yescharacteristic
316C2×C4C23×D311yescharacteristic
317C8—61no—
318C8—61no—
319C8—61no—
320C8—61no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3#54611yescharacteristic
325C2×C3#57511yescharacteristic
326C2×C3#57811yescharacteristic
327C22—21no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22C22×#6711yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—121no—
345C4—121no—
346C4—121no—
347C4—121no—
348C4C2×#23711yescharacteristic
349C4#100411yescharacteristic
350C3#331511yescharacteristic, complemented, Sylow
351C2—41no—
352C2—41no—
353C2—42no—
354C2#580611yescharacteristic
355C2#582211yescharacteristic
356C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#547C2
socleC22×C3C22×D4
hypercenter#547D3
hypocenterC3#3315
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
2. #5541C2semidirect
3. #5239C2semidirect
4. #5239C2semidirect
5. #5233C2semidirect
6. #5232C2semidirect
7. #5090C2semidirect
8. #5082C2semidirect
9. #5052C2semidirect
10. #5051C2semidirect
11. #5045C2semidirect
12. #5044C2semidirect
13. #5018C2semidirect
14. #5011C2semidirect
15. #5009C2semidirect
350. C3#3315semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#55412
#52392
#52332
#52322
#50902
#50822
#50522
#50512
#50452
#50442
#50182
#50112
#50092
#33153