Finite Group Databasebeta

← #22201

Subgroups — #22201

Order 256.

439 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic
2#2324C211yesmaximal, characteristic
3#2323C211yesmaximal, characteristic, complemented
4#1928C211yesmaximal, characteristic, complemented
5#1925C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1893C211yesmaximal, characteristic
9#1892C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1865C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#468C418yescomplemented
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34#391C2211yescharacteristic
35#390—21no—
36#390C2211yescharacteristic
37#389—21no—
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#388—21no—
43#387—21no—
44#386—21no—
45#385—21no—
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×SD16—41no—
66C2×SD16—41no—
67C2×SD16C2×C414yes—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16C2×C414yescomplemented
71C22×C8—22no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22⋊Q8—22no—
79C22⋊Q8C2×C414yescomplemented
80#121—22no—
81#121C2×C414yes—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×D4—41no—
85C4×D4—41no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—41no—
115#107C2×C414yescomplemented
116#106—41no—
117#103—41no—
118#103—41no—
119#103C2×C414yescomplemented
120#102—41no—
121#102—41no—
122#102C2×C414yes—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131C24—21no—
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8C22×C412yes—
139C2×D4—24no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—81no—
146C2×D4—81no—
147C2×D4C22×C412yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4C22×C412yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4#4112yes—
169C22×C4C2411yescharacteristic
170SD16—44no—
171M16—44no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—44no—
183C2×C8C22×C412yes—
184C2×C8C22×C412yes—
185C4⋊C4—24no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4C22×C412yes—
202C4⋊C4C22×C412yes—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—42no—
217C42—81no—
218C42—81no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C4×D412yes—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238D4—22no—
239D4—22no—
240D4—41no—
241D4—41no—
242D4—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
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×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—84no—
283C2×C4C4×D412yes—
284C2×C4C23×C411yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C8—22no—
291C8—22no—
292C8—41no—
293C8—41no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22C2×C4×D411yescharacteristic
309C22#53411yescharacteristic
310C22#53711yescharacteristic
311C22#57611yescharacteristic
312C22#57711yescharacteristic
313C22#57711yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#51811yescharacteristic
328C4#52011yescharacteristic
329C4#53811yescharacteristic
330C4#53911yescharacteristic
331C2—42no—
332C2—82no—
333C2#232511yescharacteristic
334C2#294611yescharacteristic
335C2#294711yescharacteristic
336C2#320011yescharacteristic
337C2#320311yescharacteristic
338C2#325711yescharacteristic
339C2#325811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2323C2semidirect
4. #1928C2semidirect
5. #1925C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
9. #1892C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1865C2semidirect
13. #1864C2semidirect
14. #1809C2semidirect
15. #1729C2semidirect
24. #468C4semidirect
70. C2×M16C2×C4semidirect
79. C22⋊Q8C2×C4semidirect
115. #107C2×C4semidirect
119. #103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
#23242
#23232
#19282
#19252
#19212
#19182
#18932
#18922
#18802
#18772
#18652
#18642
#18092
#17292