Finite Group Databasebeta

← #17601

Subgroups — #17601

Order 256.

441 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3523C211yesmaximal, characteristic
2#2984C211yesmaximal, characteristic, complemented
3#2320C211yesmaximal, characteristic, complemented
4#1521C212yesmaximal
5#1518C212yesmaximal, complemented
6#1515C212yesmaximal, complemented
7#1512C212yesmaximal
8#1509C211yesmaximal, characteristic
9#1506C211yesmaximal, characteristic, complemented
10#1503C211yesmaximal, characteristic, complemented
11#1500C211yesmaximal, characteristic
12#557C411yescharacteristic, complemented
13#557C411yescharacteristic, complemented
14#557C412yescomplemented
15#556C411yescharacteristic, complemented
16#556C412yescomplemented
17#552—22no—
18#552—21no—
19D4×Q8—21no—
20D4×Q8C411yescharacteristic, complemented
21#541—21no—
22#541—22no—
23#533C2211yescharacteristic
24#533C2212yes—
25#529C2211yescharacteristic
26#529C2212yes—
27#519—21no—
28C2×C4×Q8C2211yescharacteristic
29#446—21no—
30#446—22no—
31C8×Q8—21no—
32#432C2211yescharacteristic
33#432C2212yes—
34#431C2211yescharacteristic
35#431C2212yes—
36#391—21no—
37#389—21no—
38#387—22no—
39#383C2212yes—
40#382C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#376C2211yescharacteristic
43#328C2211yescharacteristic
44#328C2211yescharacteristic
45#328C2212yes—
46#328C2212yes—
47#328C2212yes—
48#328C2212yes—
49#328C2212yes—
50#327C2211yescharacteristic
51#327C2211yescharacteristic
52#327C2212yes—
53C22×Q8—21no—
54C22×Q8D411yescharacteristic
55C4⋊Q8—21no—
56C4⋊Q8—22no—
57C4⋊Q8C2×C411yescharacteristic
58C4⋊Q8C2×C412yes—
59C4⋊Q8C2×C411yescharacteristic
60C4⋊Q8C2×C412yes—
61#126—22no—
62#126—22no—
63#126—22no—
64#126—22no—
65#126—21no—
66#125—22no—
67#125—22no—
68#125—22no—
69#125—22no—
70#125—21no—
71#125—22no—
72#125—21no—
73#125C2×C411yescharacteristic
74#125C2×C412yes—
75#124C2×C411yescharacteristic
76#124C2×C412yes—
77#123—21no—
78#123—22no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—21no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—21no—
85C22⋊Q8C2311yescharacteristic
86C22⋊Q8C2312yes—
87#121C2311yescharacteristic
88C4×Q8—21no—
89C4×Q8—22no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—22no—
93C4×Q8C2×C411yescharacteristic
94C4×Q8C2×C412yes—
95C4×Q8C2×C412yes—
96C4×Q8C2×C412yes—
97C4×Q8C2×C412yes—
98C4×Q8C2×C411yescharacteristic
99C4×Q8C2×C411yescharacteristic
100C4×Q8C2×C412yes—
101C4×Q8D411yescharacteristic
102C4×Q8D411yescharacteristic
103C4×Q8D412yes—
104C4×D4—21no—
105C4×D4—22no—
106C4×D4C2×C411yescharacteristic
107C4×D4C2×C412yes—
108#117—21no—
109#117—22no—
110#117—22no—
111#117—21no—
112#117—21no—
113#117—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4D412yes—
117C2×C4⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121C2×C42C2312yes—
122C4⋊C8—22no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C22⋊C8C2311yescharacteristic
127C22⋊C8C2311yescharacteristic
128C22⋊C8C2312yes—
129#97—22no—
130#97—21no—
131C4×C8—21no—
132C4×C8—22no—
133#95—21no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95C2311yescharacteristic
138#95C2311yescharacteristic
139#95C2312yes—
140C2×Q8—22no—
141C2×Q8—21no—
142C2×Q8—42no—
143C2×Q8—44no—
144C2×Q8C22⋊C411yescharacteristic
145C2×Q8C22⋊C411yescharacteristic
146C2×Q8C22⋊C412yes—
147C2×Q8C22×C411yescharacteristic
148C2×Q8C22×C412yes—
149C2×D4—21no—
150C2×D4C22×C411yescharacteristic
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D412yes—
160C22×C4C2411yescharacteristic
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4C22⋊C412yes—
184C4⋊C4C22⋊C411yescharacteristic
185C4⋊C4C22⋊C412yes—
186C4⋊C4C22⋊C411yescharacteristic
187C4⋊C4C22⋊C412yes—
188C4⋊C4C22⋊C412yes—
189C4⋊C4C22⋊C412yes—
190C4⋊C4C22×C411yescharacteristic
191C4⋊C4C22×C412yes—
192C4⋊C4C22×C411yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—22no—
206C42—21no—
207C42—42no—
208C42—81no—
209C42C22×C412yes—
210C42C22×C411yescharacteristic
211C42C22×C411yescharacteristic
212C42C22×C412yes—
213C42C2×D411yescharacteristic
214C42C2×D412yes—
215C42C2×D412yes—
216C42C2×D411yescharacteristic
217C23—21no—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—42no—
222Q8—41no—
223Q8—42no—
224Q8—44no—
225Q8—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4C2×C22⋊C411yescharacteristic
253C2×C4C2×C22⋊C412yes—
254C2×C4C2×C22⋊C412yes—
255C2×C4C2×C22⋊C412yes—
256C2×C4C2×C22⋊C411yescharacteristic
257C2×C4C2×C22⋊C411yescharacteristic
258C2×C4C2×C22⋊C412yes—
259C2×C4C2×C22⋊C411yescharacteristic
260C2×C4C23×C411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C8—41no—
263C8—41no—
264C8—42no—
265C22—21no—
266C22—21no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22C22×C22⋊C411yescharacteristic
271C22#57711yescharacteristic
272C22#57811yescharacteristic
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—21no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—41no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—81no—
288C4—82no—
289C4—81no—
290C4#42112yes—
291C2—21no—
292C2—21no—
293C2—81no—
294C2#288411yescharacteristic
295C2#289811yescharacteristic
296C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#121C23
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2984C2semidirect
3. #2320C2semidirect
5. #1518C2semidirect
6. #1515C2semidirect
9. #1506C2semidirect
10. #1503C2semidirect
12. #557C4semidirect
13. #557C4semidirect
14. #557C4semidirect
15. #556C4semidirect
16. #556C4semidirect
20. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#35232
#29842
#23202
#15212
#15182
#15152
#15122
#15092
#15062
#15032
#15002