Finite Group Databasebeta

← #83573

Subgroups — #83573

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#550C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic, complemented
6#5232C211yesmaximal, characteristic, complemented
7#5075C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#5041C211yesmaximal, characteristic, complemented
11#5035C211yesmaximal, characteristic, complemented
12#5034C211yesmaximal, characteristic, complemented
13#5030C211yesmaximal, characteristic, complemented
14#5022C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3317—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C3×#127—21no—
20C3×#127C2211yescharacteristic
21C3×#121—21no—
22C3×#121—21no—
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4—21no—
27C3×C4×D4C2211yescharacteristic
28C2×#216C2211yescharacteristic
29#935—21no—
30C2×#214C2211yescharacteristic
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#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic
54#810C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#552—31no—
61#550D311yescharacteristic
62#490—31no—
63#482—31no—
64#481—31no—
65#479—31no—
66#468—31no—
67#465—31no—
68#460—31no—
69#452—31no—
70#452—31no—
71#441—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×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×Dic3—21no—
94C2×Dic6C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C22⋊C4D411yescharacteristic
101C3×C22⋊C4D411yescharacteristic
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104#217—21no—
105#217—21no—
106#217—21no—
107#216—22no—
108#214—41no—
109#214—41no—
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×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134#127—21no—
135#127C2×D311yescharacteristic
136#126—61no—
137#125—31no—
138#125—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142#121—21no—
143#121—21no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8C2×D311yescharacteristic
148C4×D4—21no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#117—61no—
152C2×C4⋊C4—31no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C23×C3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8#4112yes—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—41no—
181C3×D4—42no—
182C3×D4—42no—
183C3×D4—42no—
184C3×D4—81no—
185C3×D4—81no—
186C3×D4—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—42no—
205Dic6—21no—
206Dic6—21no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—22no—
212#41—22no—
213#41—41no—
214#41—41no—
215C2×Q8—31no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4#6711yescharacteristic
224C2×D4C22×D311yescharacteristic
225C2×D4C22×D311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—61no—
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231Q16—62no—
232SD16—121no—
233SD16—121no—
234M16—121no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—121no—
240C4⋊C4—21no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4#6711yescharacteristic
251C4⋊C4C22×D311yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4#6711yescharacteristic
258C22⋊C4#6711yescharacteristic
259C42—21no—
260C42—61no—
261C42—61no—
262C42C22×D311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288C23—21no—
289C23—21no—
290C23C2×#6711yescharacteristic
291C23C2×#6711yescharacteristic
292Q8—21no—
293Q8—61no—
294Q8—61no—
295Q8#23712yes—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—81no—
305D4—81no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—122no—
320C2×C4C2×#6711yescharacteristic
321C2×C4C2×#6711yescharacteristic
322C2×C4C2×#6711yescharacteristic
323C2×C4C2×#6711yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—61no—
326C8—61no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3#54611yescharacteristic
334C2×C3#57511yescharacteristic
335C2×C3#57711yescharacteristic
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22C22×#6711yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4—121no—
359C4C2×#23711yescharacteristic
360C4#100411yescharacteristic
361C3#331711yescharacteristic, complemented, Sylow
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2#580611yescharacteristic
367C2#582211yescharacteristic
368C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#550C2
socleC22×C3C22×D4
hypercenter#550D3
hypocenterC3#3317
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

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. #5075C2semidirect
8. #5071C2semidirect
9. #5042C2semidirect
10. #5041C2semidirect
11. #5035C2semidirect
12. #5034C2semidirect
13. #5030C2semidirect
14. #5022C2semidirect
15. #5011C2semidirect
361. C3#3317semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#55412
#52392
#52332
#52322
#50752
#50712
#50422
#50412
#50352
#50342
#50302
#50222
#50112
#33173