Finite Group Databasebeta

← #36434

Subgroups — #36434

Order 256.

1543 conjugacy classes of subgroups.

The subgroup lattice has 355 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
4C22×C4×Q8C211yesmaximal, characteristic, complemented
5#2850C213yesmaximal, complemented
6#2545C213yesmaximal, complemented
7#2536C213yesmaximal, complemented
8#2429C213yesmaximal, complemented
9#2426C213yesmaximal, complemented
10#2417C213yesmaximal, complemented
11#2416C213yesmaximal, complemented
12#2351C211yesmaximal, characteristic, complemented
13#2341C211yesmaximal, characteristic, complemented
14Q8×C22⋊C4C211yesmaximal, characteristic, complemented
15C4×#121C213yesmaximal, 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—
31C2×#121C2213yes—
32C2×C4×Q8—28no—
33C2×C4×Q8C2211yescharacteristic
34C2×C4×Q8C2211yescharacteristic
35C2×C4×Q8C2211yescharacteristic
36C2×C4×Q8C2213yes—
37C2×C4×Q8C2213yes—
38C2×C4×Q8C2211yescharacteristic
39C2×C4×D4C2213yes—
40C2×C4×D4C2213yes—
41C2×C4×D4C2213yes—
42C22×C4⋊C4C2213yes—
43C22×C42C2213yes—
44#398C2216yes—
45#396C2216yes—
46#396C2216yes—
47#394C2216yes—
48#391C2213yes—
49#391C2213yes—
50#390C2213yes—
51#390C2213yes—
52#390C2213yes—
53#389C2213yes—
54#389C2213yes—
55#389C2213yes—
56#388C2216yes—
57#388C2216yes—
58#388C2213yes—
59#388C2213yes—
60#386C2213yes—
61#386C2216yes—
62#386C2211yescharacteristic
63#382C2213yes—
64#380C2213yes—
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×D4C2313yes—
83C22×D4C2313yes—
84C22×D4C2313yes—
85C22×D4C2311yescharacteristic
86C23×C4C2313yes—
87C23×C4C2313yes—
88C23×C4C2311yescharacteristic
89C23×C4C2313yes—
90C23×C4C2313yes—
91C4⋊Q8—212no—
92#124—212no—
93#124—212no—
94C22⋊Q8—212no—
95C22⋊Q8—212no—
96C22⋊Q8—212no—
97C22⋊Q8—212no—
98#121—212no—
99#121—212no—
100#121—212no—
101C4×Q8—24no—
102C4×Q8—24no—
103C4×Q8—24no—
104C4×Q8—212no—
105C4×Q8—212no—
106C4×Q8D418yescomplemented
107C4×D4—212no—
108C4×D4—212no—
109C4×D4—212no—
110C2×C4⋊C4—212no—
111C2×C4⋊C4C2313yes—
112C2×C4⋊C4C2316yes—
113C2×C4⋊C4C2313yes—
114C2×C4⋊C4C2313yes—
115C2×C4⋊C4C2313yes—
116C2×C4⋊C4C2313yes—
117C2×C4⋊C4C2313yes—
118C2×C4⋊C4C2313yes—
119C2×C4⋊C4C2316yes—
120C2×C4⋊C4C2313yes—
121C2×C4⋊C4C2313yes—
122C2×C22⋊C4C2311yescharacteristic
123C2×C22⋊C4C2311yescharacteristic
124C2×C22⋊C4C2313yes—
125C2×C22⋊C4C2313yes—
126C2×C22⋊C4C2313yes—
127C2×C22⋊C4C2313yes—
128C2×C22⋊C4C2316yes—
129C2×C22⋊C4C2316yes—
130C2×C22⋊C4C2313yes—
131C2×C22⋊C4C2313yes—
132C2×C22⋊C4C2311yescharacteristic
133C2×C22⋊C4C2313yes—
134C2×C22⋊C4C2316yes—
135C2×C42—212no—
136C2×C42C2313yes—
137C2×C42C2313yes—
138C2×C42C2313yes—
139C2×C42C2313yes—
140C2×C42C2313yes—
141C2×C42C2313yes—
142#95C2311yescharacteristic
143#95C2313yes—
144#95C2313yes—
145#95C2313yes—
146#95C2311yescharacteristic
147#95C2313yes—
148#95C2316yes—
149#95C2316yes—
150C24C2413yes—
151C24C2411yescharacteristic
152C24C2411yescharacteristic
153#41—416no—
154C2×Q8—232no—
155C2×Q8—26no—
156C2×Q8—26no—
157C2×Q8—26no—
158C2×Q8—26no—
159C2×Q8—26no—
160C2×Q8—416no—
161C2×Q8C2×D4112yes—
162C2×Q8C2×D414yes—
163C2×Q8C2×D414yes—
164C2×Q8#4114yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—26no—
168C2×D4—26no—
169C2×D4—26no—
170C2×D4—26no—
171C2×D4—26no—
172C2×D4—26no—
173C2×D4—26no—
174C2×D4—26no—
175C2×D4—43no—
176C2×D4#4114yescomplemented
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—26no—
180C22×C4—26no—
181C22×C4—22no—
182C22×C4—26no—
183C22×C4—26no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—26no—
187C22×C4—26no—
188C22×C4—26no—
189C22×C4—26no—
190C22×C4—43no—
191C22×C4C2×D4112yes—
192C22×C4C2413yes—
193C22×C4C2411yescharacteristic
194C22×C4C2413yes—
195C22×C4C2413yes—
196C22×C4C2413yes—
197C22×C4C2411yescharacteristic
198C22×C4C2411yescharacteristic
199C22×C4C2413yes—
200C22×C4C2413yes—
201C22×C4C2411yescharacteristic
202C22×C4C2411yescharacteristic
203C22×C4C2413yes—
204C4⋊C4—212no—
205C4⋊C4—26no—
206C4⋊C4—26no—
207C4⋊C4—26no—
208C4⋊C4—26no—
209C4⋊C4—26no—
210C4⋊C4—26no—
211C4⋊C4—212no—
212C4⋊C4—26no—
213C4⋊C4—43no—
214C4⋊C4C2×D4112yes—
215C22⋊C4—22no—
216C22⋊C4—26no—
217C22⋊C4—26no—
218C22⋊C4—22no—
219C22⋊C4—26no—
220C22⋊C4—26no—
221C22⋊C4—26no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—26no—
225C22⋊C4—22no—
226C22⋊C4—26no—
227C22⋊C4—46no—
228C42—26no—
229C42—26no—
230C42—26no—
231C42—26no—
232C42—26no—
233C42C2×D4112yes—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—23no—
245C23—23no—
246C23—23no—
247C23—23no—
248C23—23no—
249C23—23no—
250C23—23no—
251C23C22×D412yes—
252C23C2×#4112yes—
253C23C2×#4112yes—
254C232−1+412yes—
255C23C2511yescharacteristic
256Q8—224no—
257Q8—412no—
258Q8#121116yescomplemented
259D4—28no—
260D4—412no—
261D4—412no—
262D4—412no—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—23no—
267C2×C4—23no—
268C2×C4—23no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—23no—
274C2×C4—224no—
275C2×C4—21no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—23no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—23no—
283C2×C4—23no—
284C2×C4—23no—
285C2×C4—21no—
286C2×C4—23no—
287C2×C4—21no—
288C2×C4—23no—
289C2×C4—23no—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—21no—
293C2×C4—412no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C22×D412yes—
297C2×C4C22×D416yes—
298C2×C4C22×D416yes—
299C2×C4C22×D416yes—
300C2×C4C22×D416yes—
301C2×C4C2×#4112yes—
302C2×C4C2×#4116yes—
303C2×C42−1+412yes—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—43no—
318C22—43no—
319C22—43no—
320C22—43no—
321C22—43no—
322C22—43no—
323C22—43no—
324C22#54814yescomplemented
325C22C23×D411yescharacteristic
326C22C23×D411yescharacteristic
327C22C22×#4111yescharacteristic
328C22C22×#4111yescharacteristic
329C22C2×2−1+411yescharacteristic
330C22C2×2−1+411yescharacteristic
331C22#58511yescharacteristic
332C4—26no—
333C4—22no—
334C4—26no—
335C4—22no—
336C4—22no—
337C4—41no—
338C4—43no—
339C4—43no—
340C4—43no—
341C4—41no—
342C4—43no—
343C4C2×#121112yes—
344C2—24no—
345C2—42no—
346C2—83no—
347C2C22×#12111yescharacteristic
348C2C2×#54811yescharacteristic
349C2D4×#4111yescharacteristic
350C2#348211yescharacteristic
351C2#348611yescharacteristic
352C2#349711yescharacteristic
353C2#351611yescharacteristic
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. C22×C4×Q8C2semidirect
5. #2850C2semidirect
6. #2545C2semidirect
7. #2536C2semidirect
8. #2429C2semidirect
9. #2426C2semidirect
10. #2417C2semidirect
11. #2416C2semidirect
12. #2351C2semidirect
13. #2341C2semidirect
14. Q8×C22⋊C4C2semidirect
15. C4×#121C2semidirect
78. C22×Q8D4semidirect
106. C4×Q8D4semidirect
176. C2×D4#41semidirect
258. Q8#121semidirect
324. C22#548semidirect

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
C22×C4×Q82
#28502
#25452
#25362
#24292
#24262
#24172
#24162
#23512
#23412
Q8×C22⋊C42
C4×#1212