Finite Group Databasebeta

← #35831

Subgroups — #35831

Order 256.

1574 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#550C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3C2×#548C211yesmaximal, characteristic, complemented
4C2×#520C211yesmaximal, characteristic, complemented
5#2620C213yesmaximal, complemented
6#2544C213yesmaximal, complemented
7#2537C213yesmaximal, complemented
8#2428C213yesmaximal, complemented
9#2427C213yesmaximal, complemented
10#2417C213yesmaximal, complemented
11#2416C213yesmaximal, complemented
12#2392C213yesmaximal, complemented
13Q8×C4⋊C4C211yesmaximal, characteristic, complemented
14#2341C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19#550—28no—
20#548—28no—
21#548—28no—
22C2×#127C2213yes—
23C2×#124C2213yes—
24C2×#124C2213yes—
25C2×C22⋊Q8C2213yes—
26C2×C22⋊Q8C2213yes—
27C2×#121C2213yes—
28C2×#121C2213yes—
29C2×#121C2213yes—
30C2×#121C2213yes—
31C2×#121C2216yes—
32#520—28no—
33C2×C4×Q8C2211yescharacteristic
34C2×C4×Q8C2211yescharacteristic
35C2×C4×Q8C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×D4C2213yes—
38C2×C4×D4C2213yes—
39C2×C4×D4C2213yes—
40C2×C4×D4C2213yes—
41C2×C4×D4C2213yes—
42C2×#117C2213yes—
43C22×C4⋊C4C2213yes—
44#398C2213yes—
45#397C2213yes—
46#396C2213yes—
47#396C2216yes—
48#396C2211yescharacteristic
49#394C2216yes—
50#394C2216yes—
51#394C2213yes—
52#394C2213yes—
53#394C2213yes—
54#392C2213yes—
55#391C2213yes—
56#391C2213yes—
57#390C2213yes—
58#390C2213yes—
59#389C2213yes—
60#389C2213yes—
61#386C2216yes—
62#386C2216yes—
63#384C2213yes—
64#380C2213yes—
65#380C2213yes—
66#380C2216yes—
67C4×C4⋊C4C2213yes—
68C4×C22⋊C4C2213yes—
69C4×C22⋊C4C2213yes—
70C2×#41—24no—
71C2×#41—24no—
72C2×#41—24no—
73C2×#41—24no—
74C2×#41—24no—
75C2×#41D418yescomplemented
76C22×Q8—24no—
77C22×Q8—24no—
78C22×Q8—24no—
79C22×Q8C2313yes—
80C22×Q8C2311yescharacteristic
81C22×D4C2313yes—
82C22×D4C2313yes—
83C22×D4C2313yes—
84C22×D4C2313yes—
85C22×D4C2313yes—
86C22×D4C2316yes—
87C23×C4C2313yes—
88C23×C4C2313yes—
89C23×C4C2313yes—
90C23×C4C2316yes—
91#127—212no—
92#124—212no—
93#124—212no—
94C22⋊Q8—212no—
95C22⋊Q8—212no—
96#121—212no—
97#121—212no—
98#121—212no—
99#121—212no—
100#121—412no—
101C4×Q8—24no—
102C4×Q8—24no—
103C4×Q8—24no—
104C4×Q8D418yescomplemented
105C4×D4—212no—
106C4×D4—212no—
107C4×D4—212no—
108C4×D4—212no—
109C4×D4—212no—
110#117—212no—
111C2×C4⋊C4—212no—
112C2×C4⋊C4C2313yes—
113C2×C4⋊C4C2313yes—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2313yes—
117C2×C4⋊C4C2313yes—
118C2×C4⋊C4C2313yes—
119C2×C4⋊C4C2313yes—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2316yes—
122C2×C4⋊C4C2313yes—
123C2×C4⋊C4C2313yes—
124C2×C22⋊C4C2313yes—
125C2×C22⋊C4C2313yes—
126C2×C22⋊C4C2311yescharacteristic
127C2×C22⋊C4C2311yescharacteristic
128C2×C22⋊C4C2313yes—
129C2×C22⋊C4C2313yes—
130C2×C22⋊C4C2313yes—
131C2×C22⋊C4C2313yes—
132C2×C22⋊C4C2313yes—
133C2×C22⋊C4C2316yes—
134C2×C22⋊C4C2316yes—
135C2×C22⋊C4C2311yescharacteristic
136C2×C22⋊C4C2316yes—
137C2×C22⋊C4C2313yes—
138C2×C22⋊C4C2313yes—
139C2×C42C2313yes—
140C2×C42C2313yes—
141C2×C42C2313yes—
142C2×C42C2313yes—
143#95C2311yescharacteristic
144#95C2313yes—
145#95C2313yes—
146#95C2313yes—
147#95C2313yes—
148#95C2313yes—
149#95C2316yes—
150C24C2413yes—
151C24C2413yes—
152C24C2411yescharacteristic
153#41—232no—
154#41—416no—
155C2×Q8—26no—
156C2×Q8—26no—
157C2×Q8—26no—
158C2×Q8—416no—
159C2×Q8C2×D414yes—
160C2×Q8C2×D414yes—
161C2×Q8#4114yes—
162C2×D4—26no—
163C2×D4—26no—
164C2×D4—26no—
165C2×D4—26no—
166C2×D4—212no—
167C2×D4—26no—
168C2×D4—26no—
169C2×D4—26no—
170C2×D4—26no—
171C2×D4—26no—
172C2×D4—26no—
173C2×D4—26no—
174C2×D4—43no—
175C2×D4—43no—
176C2×D4—43no—
177C2×D4—46no—
178C2×D4—46no—
179C2×D4C2×D4112yes—
180C22×C4—26no—
181C22×C4—26no—
182C22×C4—26no—
183C22×C4—26no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—212no—
189C22×C4—46no—
190C22×C4—46no—
191C22×C4C2×D4112yes—
192C22×C4C2411yescharacteristic
193C22×C4C2411yescharacteristic
194C22×C4C2413yes—
195C22×C4C2413yes—
196C22×C4C2413yes—
197C22×C4C2411yescharacteristic
198C22×C4C2413yes—
199C22×C4C2413yes—
200C22×C4C2411yescharacteristic
201C22×C4C2411yescharacteristic
202C22×C4C2411yescharacteristic
203C22×C4C2413yes—
204C4⋊C4—26no—
205C4⋊C4—26no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—26no—
209C4⋊C4—26no—
210C4⋊C4—26no—
211C4⋊C4—26no—
212C4⋊C4—22no—
213C4⋊C4—43no—
214C4⋊C4—46no—
215C4⋊C4C2×D4112yes—
216C22⋊C4—26no—
217C22⋊C4—26no—
218C22⋊C4—212no—
219C22⋊C4—26no—
220C22⋊C4—26no—
221C22⋊C4—26no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—26no—
225C22⋊C4—26no—
226C22⋊C4—212no—
227C22⋊C4—22no—
228C22⋊C4—43no—
229C22⋊C4—43no—
230C22⋊C4—46no—
231C42—26no—
232C42—26no—
233C42—26no—
234C42C2×D4112yes—
235C23—23no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—23no—
240C23—23no—
241C23—21no—
242C23—23no—
243C23—23no—
244C23—21no—
245C23—21no—
246C23—23no—
247C23—23no—
248C23—23no—
249C23—23no—
250C23—23no—
251C23—23no—
252C23—23no—
253C23—23no—
254C23C22×D416yes—
255C23H(5,𝔽2)12yes—
256C23C2511yescharacteristic
257Q8—412no—
258Q8#121116yescomplemented
259D4—224no—
260D4—412no—
261D4—412no—
262D4—412no—
263D4—412no—
264D4—812no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—23no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—23no—
272C2×C4—21no—
273C2×C4—224no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—21no—
280C2×C4—23no—
281C2×C4—23no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—23no—
293C2×C4—23no—
294C2×C4—23no—
295C2×C4—412no—
296C2×C4—412no—
297C2×C4—812no—
298C2×C4C22×D412yes—
299C2×C4C22×D416yes—
300C2×C4C22×D412yes—
301C2×C4C22×D416yes—
302C2×C4C22×D416yes—
303C2×C4C2×#4116yes—
304C2×C42−1+412yes—
305C22—26no—
306C22—22no—
307C22—22no—
308C22—26no—
309C22—26no—
310C22—22no—
311C22—41no—
312C22—43no—
313C22—41no—
314C22—43no—
315C22—41no—
316C22—43no—
317C22—41no—
318C22—43no—
319C22—43no—
320C22—43no—
321C22—43no—
322C22—43no—
323C22—43no—
324C22—43no—
325C22—43no—
326C22C23×D411yescharacteristic
327C22C23×D411yescharacteristic
328C22C22×#4111yescharacteristic
329C22C2×H(5,𝔽2)11yescharacteristic
330C22C2×2−1+411yescharacteristic
331C22#58511yescharacteristic
332C22#58511yescharacteristic
333C4—22no—
334C4—26no—
335C4—22no—
336C4—22no—
337C4—41no—
338C4—43no—
339C4—43no—
340C4—43no—
341C4—43no—
342C4—41no—
343C4—41no—
344C4C2×#121112yes—
345C2—46no—
346C2—42no—
347C2—83no—
348C2C22×#12111yescharacteristic
349C2#348811yescharacteristic
350C2#348811yescharacteristic
351C2#348911yescharacteristic
352C2#349711yescharacteristic
353C2#352211yescharacteristic
354C2#352411yescharacteristic
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×#550C2semidirect
2. C2×#548C2semidirect
3. C2×#548C2semidirect
4. C2×#520C2semidirect
5. #2620C2semidirect
6. #2544C2semidirect
7. #2537C2semidirect
8. #2428C2semidirect
9. #2427C2semidirect
10. #2417C2semidirect
11. #2416C2semidirect
12. #2392C2semidirect
13. Q8×C4⋊C4C2semidirect
14. #2341C2semidirect
15. Q8×C22⋊C4C2semidirect
75. C2×#41D4semidirect
104. C4×Q8D4semidirect
258. Q8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#5482
C2×#5202
#26202
#25442
#25372
#24282
#24272
#24172
#24162
#23922
Q8×C4⋊C42
#23412
Q8×C22⋊C42