Finite Group Databasebeta

← #30354

Subgroups — #30354

Order 256.

352 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2644C211yesmaximal, characteristic, complemented
2#2640C211yesmaximal, characteristic, complemented
3#2066C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic, complemented
9#2023C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic
11#1972C211yesmaximal, characteristic
12#1968C211yesmaximal, characteristic, complemented
13#1935C211yesmaximal, characteristic, complemented
14#1934C211yesmaximal, characteristic, complemented
15#1768C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#397—21no—
35#397—21no—
36#397—21no—
37#397—21no—
38#397—21no—
39#396—21no—
40#393—21no—
41#393—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52#376C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C2×SD16—22no—
64C2×SD16—41no—
65C2×SD16—41no—
66C2×D8—22no—
67C2×D8—41no—
68C2×D8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75#124—41no—
76#124—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#106—41no—
102#106—41no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105#103—22no—
106#103—41no—
107#103—41no—
108#102—22no—
109#102—41no—
110#102—41no—
111#102—41no—
112#102—41no—
113#97—41no—
114C4×C8—41no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—42no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162SD16—44no—
163D8—44no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8#4112yes—
175C2×C8#4112yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—84no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—22no—
279C8—22no—
280C8—41no—
281C8—41no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#53211yescharacteristic
297C22#53511yescharacteristic
298C22#53711yescharacteristic
299C22#57611yescharacteristic
300C22#57711yescharacteristic
301C22#57711yescharacteristic
302C22#57711yescharacteristic
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4#53811yescharacteristic
316C4#53811yescharacteristic
317C4#54211yescharacteristic
318C4#54211yescharacteristic
319C2—161no—
320C2—161no—
321C2#266211yescharacteristic
322C2#317311yescharacteristic
323C2#317911yescharacteristic
324C2#322111yescharacteristic
325C2#322411yescharacteristic
326C2#325711yescharacteristic
327C2#326311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2644C2semidirect
2. #2640C2semidirect
3. #2066C2semidirect
4. #2065C2semidirect
5. #2058C2semidirect
6. #2055C2semidirect
8. #2042C2semidirect
9. #2023C2semidirect
12. #1968C2semidirect
13. #1935C2semidirect
14. #1934C2semidirect
15. #1768C2semidirect

Maximal subgroups

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

subgroupindex
#26442
#26402
#20662
#20652
#20582
#20552
#20542
#20422
#20232
#20122
#19722
#19682
#19352
#19342
#17682