Finite Group Databasebeta

← #30773

Subgroups — #30773

Order 256.

422 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#500C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic
4#2440C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic
10#2023C211yesmaximal, characteristic, complemented
11#1947C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic, complemented
14#1918C211yesmaximal, characteristic, complemented
15#1852C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124—21no—
19C2×C22⋊Q8—21no—
20C2×#121C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#500—24no—
23#480—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#395—21no—
40#394—21no—
41#393—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#385—21no—
48#384C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75#124—44no—
76C22⋊Q8—44no—
77#121—22no—
78#121—41no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#107—22no—
110#107—22no—
111#107—41no—
112#107D414yes—
113C4⋊C8—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—41no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C4×C8—22no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24—21no—
129C24C2×D411yescharacteristic
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—81no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
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—42no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8C2×D412yes—
178C2×C8#4112yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×D412yes—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4D814yes—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23C2×D812yes—
228C23C22×D411yescharacteristic
229Q8—84no—
230Q8—84no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—84no—
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—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
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×C4—41no—
273C2×C4—84no—
274C2×C4C2×D812yes—
275C2×C4C2×D812yes—
276C2×C4#13712yes—
277C2×C4C22×D411yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C2×C42−1+411yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—22no—
284C8—22no—
285C8—41no—
286C8—41no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22#53611yescharacteristic
302C22D4211yescharacteristic
303C22#54711yescharacteristic
304C22C22×D811yescharacteristic
305C22C2×#13711yescharacteristic
306C22#57611yescharacteristic
307C22#57811yescharacteristic
308C4—22no—
309C4—22no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4#53611yescharacteristic
321C4#54211yescharacteristic
322C4#54811yescharacteristic
323C4#56311yescharacteristic
324C2—42no—
325C2—161no—
326C2#267411yescharacteristic
327C2#322611yescharacteristic
328C2#324311yescharacteristic
329C2#327011yescharacteristic
330C2D4×D811yescharacteristic
331C2#333411yescharacteristic
332C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#500C2semidirect
4. #2440C2semidirect
5. #2093C2semidirect
6. #2075C2semidirect
7. #2067C2semidirect
8. #2057C2semidirect
10. #2023C2semidirect
11. #1947C2semidirect
12. #1934C2semidirect
13. #1923C2semidirect
14. #1918C2semidirect
15. #1852C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4802
#24912
#24402
#20932
#20752
#20672
#20572
#20332
#20232
#19472
#19342
#19232
#19182
#18522