Finite Group Databasebeta

← #11882233

Subgroups — #11882233

Order 1152.

1176 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#532C211yesmaximal, characteristic
2#10595941C211yesmaximal, characteristic, complemented
3#10592793C212yesmaximal, complemented
4#10592774C212yesmaximal, complemented
5#10592740C214yesmaximal, complemented
6#10592718C212yesmaximal, complemented
7#10592707C212yesmaximal, complemented
8C2×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
9#83320—34nomaximal
10C2×C32×#41—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C32×#127C2211yescharacteristic
13C32×#124—21no—
14C32×#121—21no—
15C32×#121C2212yes—
16C32×C4×D4—21no—
17C32×C4×D4C2212yes—
18C2×C32×C42C2211yescharacteristic
19C2×#3754C2214yes—
20#64135—21no—
21#64133—21no—
22#64133C2212yes—
23C22×C3⋊(C3⋊C8)C2212yes—
24#64079—21no—
25#64078C2211yescharacteristic
26#64078C2211yescharacteristic
27C4×#634—21no—
28C4×#634C2212yes—
29#63659C2214yes—
30#63637C2214yes—
31#63635C2214yes—
32C4⋊(C3⋊(C3⋊C8))C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2212yes—
34C3×#532D314yes—
35#5470—34no—
36#5094—38no—
37#5075—38no—
38#5041—316no—
39#5019—38no—
40#5008—38no—
41C2×C4⋊(C3⋊C8)—34no—
42C32×#41—44no—
43C2×C32×Q8—21no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4C2312yes—
48C22×C32×C4—22no—
49C22×C32×C4D411yescharacteristic
50C22×C32×C4D411yescharacteristic
51C22×C32×C4C2311yescharacteristic
52C2×#634—21no—
53C2×#634C2312yes—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2312yes—
56C32×C22⋊C4—22no—
57C32×C22⋊C4—22no—
58C32×C42D411yescharacteristic
59C32×C42D411yescharacteristic
60C32×C42C2311yescharacteristic
61C32×C42C2311yescharacteristic
62#3757—22no—
63#3754—44no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)—21no—
67C4⋊(C3⋊Dic3)C2312yes—
68#3750—22no—
69#3750—22no—
70C4×C3⋊Dic3—22no—
71C2×C3⋊(C3⋊C8)—42no—
72C2×C3⋊(C3⋊C8)C2314yes—
73#3246—91noSylow
74C2×C3×#41—24no—
75C3×C4⋊Q8C2×D314yes—
76C3×#127C2×D314yes—
77C3×#124—24no—
78C3×#121—24no—
79C3×#121C2×D318yes—
80C3×C4×D4—24no—
81C3×C4×D4C2×D318yes—
82C2×C3×C42C2×D314yes—
83C2×#214—316no—
84#926—64no—
85#924—38no—
86#924—64no—
87C22×C3⋊C8—38no—
88#870—64no—
89#869—34no—
90#869—34no—
91C4×Dic6—38no—
92C4×Dic6—64no—
93#832—316no—
94#810—316no—
95#808—316no—
96C4⋊(C3⋊C8)—38no—
97C4⋊(C3⋊C8)—38no—
98C23×C32—22no—
99C23×C32C2×D411yescharacteristic
100C32×Q8—42no—
101C32×D4—22no—
102C32×D4—22no—
103C32×D4—42no—
104C32×D4—44no—
105C32×D4—42no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—41no—
110C2×C32×C4—44no—
111C2×C32×C4SD1614yescomplemented
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2411yescharacteristic
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120#634—22no—
121#634—22no—
122#634—42no—
123C3⋊(C3⋊C8)—24no—
124#533—91no—
125#532C3⋊D311yescharacteristic
126#476—92no—
127#475—92no—
128#465—94no—
129#461—92no—
130#460—92no—
131C2×C4⋊C8—91no—
132C3×#41—416no—
133C2×C3×Q8—24no—
134C2×C3×D4—28no—
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4C22×D318yes—
138C22×C3×C4—28no—
139C22×C3×C4#6714yes—
140C22×C3×C4#6714yes—
141C22×C3×C4C22×D314yes—
142C2×Dic6—38no—
143C2×Dic6—64no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4C22×D318yes—
146C3×C22⋊C4—28no—
147C3×C22⋊C4—28no—
148C3×C42#6714yes—
149C3×C42#6714yes—
150C3×C42C22×D314yes—
151C3×C42C22×D314yes—
152#217—68no—
153#214—1216no—
154C4⋊Dic3—38no—
155C4⋊Dic3—64no—
156C4⋊Dic3—64no—
157C4⋊Dic3—64no—
158Dic3⋊C4—68no—
159Dic3⋊C4—68no—
160C4×Dic3—68no—
161C2×C3⋊C8—316no—
162C2×C3⋊C8—128no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32C2×SD1612yes—
169C22×C32C22×D411yescharacteristic
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—42no—
174C32×C4C2×SD1612yes—
175C32×C4C2×SD1612yes—
176C32×C4H(5,𝔽2)11yescharacteristic
177C32×C42−1+411yescharacteristic
178C3⋊Dic3—42no—
179C3⋊Dic3—42no—
180C2×#41—21no—
181C2×SD16—94no—
182C22×C8—92no—
183C4⋊Q8—91no—
184C4⋊Q8—91no—
185C4⋊Q8C2×C3⋊D311yescharacteristic
186#127C2×C3⋊D311yescharacteristic
187#125—181no—
188#124—21no—
189C22⋊Q8—92no—
190C22⋊Q8—181no—
191#121—21no—
192#121C2×C3⋊D312yes—
193C4×Q8—92no—
194C4×Q8—181no—
195C4×D4—21no—
196C4×D4C2×C3⋊D312yes—
197#117—181no—
198C2×C42C2×C3⋊D311yescharacteristic
199#106—94no—
200C4⋊C8—92no—
201C4⋊C8—92no—
202#103—94no—
203#102—94no—
204C23×C3—28no—
205C23×C3C2×#6714yes—
206C3×Q8—48no—
207C3×D4—28no—
208C3×D4—28no—
209C3×D4—48no—
210C3×D4—416no—
211C3×D4—48no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—28no—
215C2×C3×C4—44no—
216C2×C3×C4—416no—
217C2×C3×C4#214116yescomplemented
218C2×C3×C4C2×#6714yes—
219C2×C3×C4C2×#6714yes—
220C2×C3×C4C2×#6714yes—
221C2×C3×C4C2×#6714yes—
222C2×C3×C4C2×#6714yes—
223C2×C3×C4C23×D314yes—
224C2×Dic3—68no—
225C2×Dic3—68no—
226Dic6—68no—
227Dic6—68no—
228Dic6—128no—
229C3⋊C8—616no—
230C2×C32—22no—
231C2×C32—82no—
232C2×C32#53711yescharacteristic
233C2×C32C22×SD1611yescharacteristic
234C2×C32#57511yescharacteristic
235#41—44no—
236C2×Q8—21no—
237C2×Q8—92no—
238C2×Q8—181no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4C22×C3⋊D312yes—
243C22×C4—22no—
244C22×C4#63811yescharacteristic
245C22×C4#63811yescharacteristic
246C22×C4C22×C3⋊D311yescharacteristic
247SD16—364no—
248C2×C8—94no—
249C2×C8—362no—
250C4⋊C4—21no—
251C4⋊C4—92no—
252C4⋊C4—181no—
253C4⋊C4—181no—
254C4⋊C4—182no—
255C4⋊C4—182no—
256C4⋊C4—181no—
257C4⋊C4C22×C3⋊D312yes—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—182no—
261C42—182no—
262C42#63811yescharacteristic
263C42#63811yescharacteristic
264C42C22×C3⋊D311yescharacteristic
265C42C22×C3⋊D311yescharacteristic
266C22×C3—24no—
267C22×C3—24no—
268C22×C3—48no—
269C22×C3—48no—
270C22×C3—48no—
271C22×C3C2×#21418yes—
272C22×C3C22×#6714yes—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—24no—
276C3×C4—48no—
277C3×C4C2×#21418yes—
278C3×C4C2×#21418yes—
279C3×C4#100914yes—
280C3×C4#101014yes—
281Dic3—128no—
282Dic3—128no—
283C32#324611yescharacteristic, complemented, Sylow
284C23—22no—
285C23C2×#63811yescharacteristic
286Q8—42no—
287Q8—182no—
288Q8—182no—
289Q8—362no—
290D4—22no—
291D4—22no—
292D4—42no—
293D4—44no—
294D4—42no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4#375414yescomplemented
303C2×C4C2×#63811yescharacteristic
304C2×C4C2×#63811yescharacteristic
305C2×C4C2×#63811yescharacteristic
306C2×C4C2×#63811yescharacteristic
307C2×C4C2×#63811yescharacteristic
308C2×C4C23×C3⋊D311yescharacteristic
309C8—184no—
310C2×C3—28no—
311C2×C3—88no—
312C2×C3C22×#21414yes—
313C2×C3#582214yes—
314C2×C3#583314yes—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22C2×#375412yes—
321C22C22×#63811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—362no—
327C4—362no—
328C4C2×#375412yes—
329C4C2×#375412yes—
330C4#6436611yescharacteristic
331C4#6436711yescharacteristic
332C3#8332014yescomplemented
333C2—22no—
334C2—82no—
335C2C22×#375411yescharacteristic
336C2#1059629311yescharacteristic
337C2#1059630411yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC32×#532C2
socleC22×C32C22×D4
hypercenter#532C3⋊D3
hypocenterC32#3246
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
2. #10595941C2semidirect
3. #10592793C2semidirect
4. #10592774C2semidirect
5. #10592740C2semidirect
6. #10592718C2semidirect
7. #10592707C2semidirect
8. C2×C4⋊(C3⋊(C3⋊C8))C2semidirect
111. C2×C32×C4SD16semidirect
217. C2×C3×C4#214semidirect
283. C32#3246semidirect
302. C2×C4#3754semidirect
332. C3#83320semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
#105959412
#105927932
#105927742
#105927402
#105927182
#105927072
C2×C4⋊(C3⋊(C3⋊C8))2
#833203