Finite Group Databasebeta

← #30699

Subgroups — #30699

Order 256.

342 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3137C211yesmaximal, characteristic, complemented
2#3098C211yesmaximal, characteristic
3#3083C211yesmaximal, characteristic, complemented
4#3063C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2064C211yesmaximal, characteristic, complemented
7#2060C211yesmaximal, characteristic, complemented
8#2052C211yesmaximal, characteristic, complemented
9#2039C211yesmaximal, characteristic
10#2038C211yesmaximal, characteristic, complemented
11#1952C211yesmaximal, characteristic, complemented
12#1929C211yesmaximal, characteristic, complemented
13#1882C211yesmaximal, characteristic, complemented
14#1866C211yesmaximal, characteristic, complemented
15#1816C211yesmaximal, characteristic, complemented
16C2×#137C2211yescharacteristic
17C2×#131C2211yescharacteristic
18#544C2211yescharacteristic
19#537C2211yescharacteristic
20#536C2211yescharacteristic
21#520C2211yescharacteristic
22#491—21no—
23#489—21no—
24#488—21no—
25#487—21no—
26#485—21no—
27#484—21no—
28#482—21no—
29#480—21no—
30#479—21no—
31#478—21no—
32#477—21no—
33#474—21no—
34#470—21no—
35#468—21no—
36#467—21no—
37#465—21no—
38C2×#108C2211yescharacteristic
39C2×#108C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#428C2211yescharacteristic
43C2×#107C2211yescharacteristic
44C2×#107C2211yescharacteristic
45#424C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#419C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#416C2211yescharacteristic
53#416C2211yescharacteristic
54C2×#103C2211yescharacteristic
55C2×#103C2211yescharacteristic
56C2×#101C2211yescharacteristic
57#408C2211yescharacteristic
58#405C2211yescharacteristic
59#391C2211yescharacteristic
60#384C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70#137—22no—
71#137—41no—
72C2×Q16—21no—
73C2×SD16—21no—
74#131—22no—
75#131—41no—
76C2×M16—21no—
77C2×M16C2311yescharacteristic
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C22×C8—21no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87#125—21no—
88#125—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98#121—41no—
99C4×Q8—21no—
100C4×D4—21no—
101C4×D4—41no—
102#117—21no—
103#117C2311yescharacteristic
104#117C2311yescharacteristic
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#108—22no—
112#108—41no—
113#107—21no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—21no—
118#107—21no—
119#107—21no—
120#106—21no—
121#106—21no—
122#106—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125#104—41no—
126#104—41no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103D411yescharacteristic
137#103D411yescharacteristic
138#103D411yescharacteristic
139#103D411yescharacteristic
140#102—21no—
141#102—21no—
142#102—21no—
143#102D411yescharacteristic
144#102D411yescharacteristic
145#102D411yescharacteristic
146#102D411yescharacteristic
147#101—41no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150#97—21no—
151C4×C8—21no—
152#95—41no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—41no—
168C2×Q8C2×D411yescharacteristic
169C2×Q8C2×D411yescharacteristic
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4C2×D411yescharacteristic
174C2×D4C2×D411yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4C2411yescharacteristic
184Q16—42no—
185SD16—42no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192M16—21no—
193M16—42no—
194M16#4112yes—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—42no—
204C2×C8C2×D411yescharacteristic
205C2×C8C2×D411yescharacteristic
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4C2×D411yescharacteristic
223C4⋊C4C2×D411yescharacteristic
224C4⋊C4C2×D411yescharacteristic
225C4⋊C4C2×D411yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C23—21no—
239C23—21no—
240C23H(5,𝔽2)11yescharacteristic
241Q8—21no—
242Q8—21no—
243Q8—21no—
244Q8—21no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—161no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—161no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—161no—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C42−1+411yescharacteristic
287C2×C42−1+411yescharacteristic
288C8—21no—
289C8—21no—
290C8—22no—
291C8—21no—
292C8—21no—
293C8—41no—
294C8—41no—
295C8—41no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22#54211yescharacteristic
305C22D4211yescharacteristic
306C22#56211yescharacteristic
307C4—42no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4#53611yescharacteristic
319C4#53711yescharacteristic
320C4#54611yescharacteristic
321C4#54711yescharacteristic
322C2—21no—
323C2—21no—
324C2—81no—
325C2—81no—
326C2#267211yescharacteristic
327C2#328811yescharacteristic
328C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3137C2semidirect
3. #3083C2semidirect
5. #2095C2semidirect
6. #2064C2semidirect
7. #2060C2semidirect
8. #2052C2semidirect
10. #2038C2semidirect
11. #1952C2semidirect
12. #1929C2semidirect
13. #1882C2semidirect
14. #1866C2semidirect
15. #1816C2semidirect

Maximal subgroups

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

subgroupindex
#31372
#30982
#30832
#30632
#20952
#20642
#20602
#20522
#20392
#20382
#19522
#19292
#18822
#18662
#18162