Finite Group Databasebeta

← #38985

Subgroups — #38985

Order 256.

909 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#2858C211yesmaximal, characteristic
2#2715C214yesmaximal, complemented
3#2700C214yesmaximal, complemented
4#2644C212yesmaximal, complemented
5#2592C212yesmaximal, complemented
6#2567C211yesmaximal, characteristic, complemented
7#2548C211yesmaximal, characteristic
8#2538C212yesmaximal, complemented
9#2512C212yesmaximal, complemented
10#2512C212yesmaximal, complemented
11#2452C212yesmaximal, complemented
12#2447C212yesmaximal, complemented
13#2369C211yesmaximal, characteristic, complemented
14#2366C211yesmaximal, characteristic
15#2319C212yesmaximal, complemented
16C4×C4⋊Q8C211yesmaximal, characteristic, complemented
17#2295C211yesmaximal, characteristic, complemented
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#126C2214yes—
22C2×#124C2212yes—
23C2×#124C2212yes—
24C2×#124C2212yes—
25C2×C22⋊Q8C2214yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2212yes—
28C2×C4×D4C2212yes—
29C2×C4×D4C2212yes—
30#397C2214yes—
31#397C2214yes—
32#397C2214yes—
33#397C2214yes—
34#396C2214yes—
35#396C2214yes—
36#396C2214yes—
37#395C2214yes—
38#394C2214yes—
39#393C2214yes—
40#391C2212yes—
41#391C2212yes—
42#391C2211yescharacteristic
43#391C2214yes—
44#391C2211yescharacteristic
45#390C2212yes—
46#390C2212yes—
47#390C2212yes—
48#390C2212yes—
49#390C2212yes—
50#390C2212yes—
51#390C2212yes—
52#390C2212yes—
53#390C2212yes—
54#389C2211yescharacteristic
55#389C2211yescharacteristic
56#389C2212yes—
57#389C2212yes—
58#389C2212yes—
59#389C2212yes—
60#389C2212yes—
61#389C2211yescharacteristic
62#389C2211yescharacteristic
63#388C2214yes—
64#388C2214yes—
65#388C2214yes—
66#387C2212yes—
67#387C2212yes—
68#386C2214yes—
69#385C2214yes—
70#384C2211yescharacteristic
71#384C2212yes—
72#381C2214yes—
73#380C2214yes—
74C4×C4⋊C4C2211yescharacteristic
75C4×C4⋊C4C2211yescharacteristic
76C4×C4⋊C4C2212yes—
77C4×C4⋊C4C2212yes—
78C4×C4⋊C4C2212yes—
79C4×C4⋊C4C2211yescharacteristic
80C4×C4⋊C4C2212yes—
81#376C2212yes—
82C43C2211yescharacteristic
83C22×Q8C2311yescharacteristic
84C22×Q8C2312yes—
85C22×Q8C2312yes—
86C22×D4C2312yes—
87C22×D4C2312yes—
88C22×D4C2312yes—
89C23×C4C2314yes—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92#127—24no—
93#126—48no—
94#124—44no—
95#124—44no—
96#124—44no—
97C22⋊Q8—48no—
98C4×Q8—24no—
99C4×Q8—44no—
100C4×D4—28no—
101C4×D4—44no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2312yes—
105C2×C4⋊C4C2312yes—
106C2×C4⋊C4C2312yes—
107C2×C4⋊C4C2312yes—
108C2×C4⋊C4C2312yes—
109C2×C4⋊C4C2312yes—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2312yes—
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2312yes—
116C2×C4⋊C4C2312yes—
117C2×C4⋊C4C2312yes—
118C2×C4⋊C4C2314yes—
119C2×C4⋊C4C2314yes—
120C2×C4⋊C4C2314yes—
121C2×C4⋊C4C2311yescharacteristic
122C2×C22⋊C4C2314yes—
123C2×C22⋊C4C2314yes—
124C2×C22⋊C4C2314yes—
125C2×C22⋊C4C2314yes—
126C2×C22⋊C4C2314yes—
127C2×C22⋊C4C2314yes—
128C2×C22⋊C4C2314yes—
129C2×C22⋊C4C2314yes—
130C2×C22⋊C4C2314yes—
131C2×C22⋊C4C2314yes—
132C2×C22⋊C4C2314yes—
133C2×C42C2312yes—
134C2×C42C2312yes—
135C2×C42C2311yescharacteristic
136C2×C42C2312yes—
137C2×C42C2312yes—
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140C2×C42C2311yescharacteristic
141C2×C42C2311yescharacteristic
142C2×C42C2311yescharacteristic
143C2×C42C2314yes—
144C2×C42C2311yescharacteristic
145#95C2312yes—
146#95C2312yes—
147#95C2312yes—
148#95C2312yes—
149#95C2314yes—
150#95C2312yes—
151#95C2312yes—
152#95C2312yes—
153#95C2312yes—
154#95C2312yes—
155#95C2312yes—
156#95C2314yes—
157#95C2312yes—
158#95C2314yes—
159#95C2314yes—
160C24C2414yes—
161C2×Q8—22no—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—24no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—42no—
178C2×D4—42no—
179C22×C4—28no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4C2411yescharacteristic
183C22×C4C2412yes—
184C22×C4C2412yes—
185C22×C4C2412yes—
186C22×C4C2411yescharacteristic
187C22×C4C2412yes—
188C22×C4C2412yes—
189C22×C4C2412yes—
190C22×C4C2414yes—
191C22×C4C2412yes—
192C22×C4C2411yescharacteristic
193C22×C4C2412yes—
194C22×C4C2411yescharacteristic
195C22×C4C2411yescharacteristic
196C22×C4C2411yescharacteristic
197C22×C4C2411yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4#4114yes—
218C22⋊C4—28no—
219C22⋊C4—28no—
220C22⋊C4—28no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C42—24no—
230C42—24no—
231C42—22no—
232C42—24no—
233C42—22no—
234C42—24no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—44no—
240C42#4114yes—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23C2511yescharacteristic
249Q8—48no—
250Q8—44no—
251Q8—84no—
252D4—48no—
253D4—48no—
254D4—84no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—88no—
294C2×C4C2×#4112yes—
295C2×C4C2×#4114yes—
296C2×C4C2×#4112yes—
297C2×C4C2×#4112yes—
298C2×C4C2×#4112yes—
299C2×C4H(5,𝔽2)12yes—
300C2×C4H(5,𝔽2)12yes—
301C2×C4H(5,𝔽2)12yes—
302C2×C42−1+412yes—
303C2×C42−1+412yes—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22C22×#4111yescharacteristic
312C22C22×#4111yescharacteristic
313C22C2×H(5,𝔽2)11yescharacteristic
314C22C2×H(5,𝔽2)11yescharacteristic
315C22C2×H(5,𝔽2)11yescharacteristic
316C22C2×2−1+411yescharacteristic
317C22C2×2−1+411yescharacteristic
318C4—22no—
319C4—24no—
320C4—22no—
321C4—22no—
322C4—42no—
323C4—42no—
324C4—44no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—41no—
332C4#54214yes—
333C4#55614yes—
334C2—84no—
335C2C2×#54211yescharacteristic
336C2C2×#55611yescharacteristic
337C2#350811yescharacteristic
338C2#351911yescharacteristic
339C2#352211yescharacteristic
340C2#354811yescharacteristic
341C2#355011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2715C2semidirect
3. #2700C2semidirect
4. #2644C2semidirect
5. #2592C2semidirect
6. #2567C2semidirect
8. #2538C2semidirect
9. #2512C2semidirect
10. #2512C2semidirect
11. #2452C2semidirect
12. #2447C2semidirect
13. #2369C2semidirect
15. #2319C2semidirect
16. C4×C4⋊Q8C2semidirect
17. #2295C2semidirect

Maximal subgroups

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

subgroupindex
#28582
#27152
#27002
#26442
#25922
#25672
#25482
#25382
#25122
#24522
#24472
#23692
#23662
#23192
C4×C4⋊Q82
#22952