Finite Group Databasebeta

← #31638

Subgroups — #31638

Order 256.

410 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2567C211yesmaximal, characteristic, complemented
2#2445C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2014C211yesmaximal, characteristic
9#1957C211yesmaximal, characteristic
10#1933C211yesmaximal, characteristic, complemented
11#1916C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#123—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#395—21no—
34#395—21no—
35#394—21no—
36#394—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389—21no—
43#389—21no—
44#389C2211yescharacteristic
45#386—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78#127—22no—
79#127—41no—
80#123—44no—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×D4—22no—
84C4×D4—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
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×C22⋊C4—21no—
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×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C4⋊C8—41no—
111#103—22no—
112#103—41no—
113#102—41no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119C4×C8—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24—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×Q8#4112yes—
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—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—81no—
154C2×D4—81no—
155C2×D4#4112yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—81no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—44no—
173SD16—44no—
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—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C4⋊C4—22no—
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—21no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4SD1614yescomplemented
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238Q8—84no—
239D4—22no—
240D4—22no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—84no—
248D4—84no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
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—84no—
282C2×C4C2×SD1612yes—
283C2×C4C2×SD1612yes—
284C2×C4C2×SD1612yes—
285C2×C4#13712yes—
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C4H(5,𝔽2)11yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C42−1+411yescharacteristic
292C8—41no—
293C8—41no—
294C8—41no—
295C8—41no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22#53511yescharacteristic
311C22#54611yescharacteristic
312C22#54811yescharacteristic
313C22C22×SD1611yescharacteristic
314C22C2×#13711yescharacteristic
315C22#57611yescharacteristic
316C22#57611yescharacteristic
317C4—22no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4#53811yescharacteristic
330C4#53911yescharacteristic
331C4#55111yescharacteristic
332C4#56211yescharacteristic
333C2—82no—
334C2—161no—
335C2#269611yescharacteristic
336C2#320611yescharacteristic
337C2#322011yescharacteristic
338C2#329511yescharacteristic
339C2#331411yescharacteristic
340C2#336211yescharacteristic
341C2#337511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2567C2semidirect
3. #2100C2semidirect
4. #2071C2semidirect
5. #2055C2semidirect
6. #2023C2semidirect
7. #2023C2semidirect
10. #1933C2semidirect
11. #1916C2semidirect
12. #1880C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. #1802C2semidirect
204. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#25672
#24452
#21002
#20712
#20552
#20232
#20142
#19572
#19332
#19162
#18802
#18772
#18022