Finite Group Databasebeta

← #35771

Subgroups — #35771

Order 256.

1474 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3C2×#548C211yesmaximal, characteristic, complemented
4C2×#519C211yesmaximal, characteristic, complemented
5#2620C213yesmaximal, complemented
6#2545C213yesmaximal, complemented
7#2538C213yesmaximal, complemented
8#2429C213yesmaximal, complemented
9#2426C213yesmaximal, complemented
10#2415C213yesmaximal, complemented
11#2414C213yesmaximal, complemented
12#2392C213yesmaximal, complemented
13Q8×C4⋊C4C211yesmaximal, characteristic, complemented
14#2341C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19D4×Q8—28no—
20#548—28no—
21#548—28no—
22C2×C4⋊Q8C2213yes—
23C2×#124C2213yes—
24C2×#124C2213yes—
25C2×C22⋊Q8C2213yes—
26C2×C22⋊Q8C2213yes—
27C2×C22⋊Q8C2213yes—
28C2×C22⋊Q8C2213yes—
29C2×#121C2213yes—
30C2×#121C2213yes—
31#519—28no—
32C2×C4×Q8C2211yescharacteristic
33C2×C4×Q8C2211yescharacteristic
34C2×C4×Q8C2211yescharacteristic
35C2×C4×Q8C2213yes—
36C2×C4×Q8C2213yes—
37C2×C4×Q8C2211yescharacteristic
38C2×C4×D4C2213yes—
39C2×C4×D4C2213yes—
40C2×C4×D4C2213yes—
41C2×#117C2213yes—
42C2×#117C2213yes—
43#398C2213yes—
44#397C2213yes—
45#396C2216yes—
46#396C2216yes—
47#396C2213yes—
48#396C2213yes—
49#394C2213yes—
50#394C2216yes—
51#394C2211yescharacteristic
52#394C2213yes—
53#392C2213yes—
54#391C2213yes—
55#391C2213yes—
56#390C2213yes—
57#390C2213yes—
58#390C2216yes—
59#389C2213yes—
60#389C2213yes—
61#389C2216yes—
62#388C2216yes—
63#388C2216yes—
64#384C2213yes—
65#380C2213yes—
66#380C2213yes—
67C4×C4⋊C4C2213yes—
68C4×C22⋊C4C2213yes—
69C4×C22⋊C4C2213yes—
70C2×#41—24no—
71C2×#41—24no—
72C2×#41—24no—
73C22×Q8—24no—
74C22×Q8—24no—
75C22×Q8—24no—
76C22×Q8—24no—
77C22×Q8—24no—
78C22×Q8D418yescomplemented
79C22×Q8C2313yes—
80C22×Q8C2313yes—
81C22×Q8C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C22×D4C2313yes—
84C22×D4C2313yes—
85C22×D4C2313yes—
86C23×C4C2313yes—
87C23×C4C2313yes—
88C23×C4C2313yes—
89C4⋊Q8—212no—
90#124—212no—
91#124—212no—
92C22⋊Q8—212no—
93C22⋊Q8—212no—
94C22⋊Q8—212no—
95C22⋊Q8—212no—
96#121—212no—
97#121—212no—
98C4×Q8—24no—
99C4×Q8—24no—
100C4×Q8—24no—
101C4×Q8—212no—
102C4×Q8—212no—
103C4×Q8D418yescomplemented
104C4×D4—212no—
105C4×D4—212no—
106C4×D4—212no—
107#117—212no—
108#117—212no—
109C2×C4⋊C4C2313yes—
110C2×C4⋊C4C2313yes—
111C2×C4⋊C4C2313yes—
112C2×C4⋊C4C2313yes—
113C2×C4⋊C4C2313yes—
114C2×C4⋊C4C2313yes—
115C2×C4⋊C4C2313yes—
116C2×C4⋊C4C2313yes—
117C2×C4⋊C4C2316yes—
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2313yes—
120C2×C4⋊C4C2316yes—
121C2×C4⋊C4C2313yes—
122C2×C22⋊C4C2313yes—
123C2×C22⋊C4C2311yescharacteristic
124C2×C22⋊C4C2311yescharacteristic
125C2×C22⋊C4C2311yescharacteristic
126C2×C22⋊C4C2313yes—
127C2×C22⋊C4C2313yes—
128C2×C22⋊C4C2313yes—
129C2×C22⋊C4C2313yes—
130C2×C22⋊C4C2316yes—
131C2×C22⋊C4C2316yes—
132C2×C22⋊C4C2311yescharacteristic
133C2×C22⋊C4C2313yes—
134C2×C22⋊C4C2313yes—
135C2×C22⋊C4C2313yes—
136C2×C22⋊C4C2316yes—
137C2×C42C2313yes—
138C2×C42C2316yes—
139C2×C42C2313yes—
140C2×C42C2313yes—
141C2×C42C2313yes—
142#95C2311yescharacteristic
143#95C2313yes—
144#95C2313yes—
145#95C2313yes—
146#95C2313yes—
147#95C2313yes—
148#95C2316yes—
149C24C2413yes—
150C24C2411yescharacteristic
151C24C2411yescharacteristic
152#41—416no—
153C2×Q8—232no—
154C2×Q8—26no—
155C2×Q8—26no—
156C2×Q8—26no—
157C2×Q8—26no—
158C2×Q8—26no—
159C2×Q8—416no—
160C2×Q8C2×D4112yes—
161C2×Q8C2×D414yes—
162C2×Q8C2×D414yes—
163C2×Q8#4114yes—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—26no—
167C2×D4—26no—
168C2×D4—26no—
169C2×D4—26no—
170C2×D4—26no—
171C2×D4—26no—
172C2×D4—22no—
173C2×D4—43no—
174C2×D4—43no—
175C2×D4—43no—
176C22×C4—26no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—26no—
180C22×C4—26no—
181C22×C4—26no—
182C22×C4—26no—
183C22×C4—26no—
184C22×C4C2×D4112yes—
185C22×C4C2411yescharacteristic
186C22×C4C2411yescharacteristic
187C22×C4C2413yes—
188C22×C4C2413yes—
189C22×C4C2413yes—
190C22×C4C2411yescharacteristic
191C22×C4C2413yes—
192C22×C4C2413yes—
193C22×C4C2411yescharacteristic
194C22×C4C2413yes—
195C22×C4C2411yescharacteristic
196C22×C4C2413yes—
197C4⋊C4—26no—
198C4⋊C4—26no—
199C4⋊C4—26no—
200C4⋊C4—26no—
201C4⋊C4—26no—
202C4⋊C4—26no—
203C4⋊C4—26no—
204C4⋊C4—26no—
205C4⋊C4—212no—
206C4⋊C4—22no—
207C4⋊C4—43no—
208C4⋊C4—46no—
209C4⋊C4C2×D4112yes—
210C22⋊C4—22no—
211C22⋊C4—26no—
212C22⋊C4—22no—
213C22⋊C4—26no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—26no—
217C22⋊C4—26no—
218C22⋊C4—26no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—26no—
222C22⋊C4—46no—
223C22⋊C4—43no—
224C22⋊C4—43no—
225C42—212no—
226C42—26no—
227C42—26no—
228C42—26no—
229C42C2×D4112yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—23no—
243C23—23no—
244C23—23no—
245C23—23no—
246C23—23no—
247C23—23no—
248C23—23no—
249C23C22×D412yes—
250C232−1+412yes—
251C23C2511yescharacteristic
252Q8—224no—
253Q8—412no—
254Q8#121116yes—
255D4—28no—
256D4—412no—
257D4—412no—
258D4—412no—
259C2×C4—21no—
260C2×C4—23no—
261C2×C4—23no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—23no—
265C2×C4—21no—
266C2×C4—23no—
267C2×C4—224no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—23no—
271C2×C4—23no—
272C2×C4—23no—
273C2×C4—23no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—23no—
277C2×C4—21no—
278C2×C4—23no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—23no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—23no—
289C2×C4—412no—
290C2×C4—412no—
291C2×C4C22×D416yes—
292C2×C4C22×D412yes—
293C2×C4C22×D416yes—
294C2×C4C22×D416yes—
295C2×C4C22×D416yes—
296C2×C4C2×#4116yes—
297C2×C42−1+412yes—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—43no—
313C22—43no—
314C22—43no—
315C22—43no—
316C22—43no—
317C22—43no—
318C22—43no—
319C22C23×D411yescharacteristic
320C22C23×D411yescharacteristic
321C22C22×#4111yescharacteristic
322C22C2×2−1+411yescharacteristic
323C22C2×2−1+411yescharacteristic
324C22#58511yescharacteristic
325C22#58511yescharacteristic
326C4—26no—
327C4—22no—
328C4—26no—
329C4—22no—
330C4—41no—
331C4—43no—
332C4—41no—
333C4—43no—
334C4—43no—
335C4—41no—
336C4—43no—
337C4C2×#121112yes—
338C2—42no—
339C2—42no—
340C2—83no—
341C2C22×#12111yescharacteristic
342C2#348611yescharacteristic
343C2#348711yescharacteristic
344C2#348911yescharacteristic
345C2#349711yescharacteristic
346C2#351611yescharacteristic
347C2#351811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×D4×Q8C2semidirect
2. C2×#548C2semidirect
3. C2×#548C2semidirect
4. C2×#519C2semidirect
5. #2620C2semidirect
6. #2545C2semidirect
7. #2538C2semidirect
8. #2429C2semidirect
9. #2426C2semidirect
10. #2415C2semidirect
11. #2414C2semidirect
12. #2392C2semidirect
13. Q8×C4⋊C4C2semidirect
14. #2341C2semidirect
15. Q8×C22⋊C4C2semidirect
78. C22×Q8D4semidirect
103. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#5482
C2×#5192
#26202
#25452
#25382
#24292
#24262
#24152
#24142
#23922
Q8×C4⋊C42
#23412
Q8×C22⋊C42