Finite Group Databasebeta

← #21463

Subgroups — #21463

Order 256.

672 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#519C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic
3C2×#441C211yesmaximal, characteristic, complemented
4C2×C4×Q16C211yesmaximal, characteristic, complemented
5#2323C211yesmaximal, characteristic
6#1878C211yesmaximal, characteristic, complemented
7#1878C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic
9#1865C211yesmaximal, characteristic, complemented
10#1863C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1787C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20#519—24no—
21#519—24no—
22C2×C4×Q8—21no—
23C2×C4×Q8—21no—
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#451C418yescomplemented
34#441—24no—
35C4×Q16—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#391C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic
53#388—21no—
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#336C2211yescharacteristic
63#336C2211yescharacteristic
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—21no—
69C22×Q8—22no—
70C22×Q8C2×C414yes—
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16C2×C414yes—
81C2×Q16C2×C414yes—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22⋊Q8—22no—
87C22⋊Q8C2×C414yes—
88C4×Q8—28no—
89C4×Q8—28no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8D414yes—
99C4×Q8D414yes—
100#117—28no—
101#117—28no—
102#117—22no—
103#117—22no—
104#117—22no—
105#117—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4D411yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—22no—
141#103—22no—
142#103C2×C414yes—
143#103C2×C414yes—
144C22⋊C8—22no—
145C22⋊C8C2×C414yes—
146#97—22no—
147C4×C8—22no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24C2×D411yescharacteristic
154C2×Q8—24no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—24no—
158C2×Q8—28no—
159C2×Q8—28no—
160C2×Q8—24no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8C22×C412yes—
173C2×Q8C22×C412yes—
174C2×Q8C22×C412yes—
175C2×Q8C2×D412yes—
176C2×Q8C2×D412yes—
177C2×Q8#4112yes—
178C2×Q8#4112yes—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4C22×C412yes—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4#4112yes—
201C22×C4C2411yescharacteristic
202Q16—28no—
203Q16—28no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8C22×C412yes—
215C2×C8C22×C412yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—81no—
232C4⋊C4C22×C412yes—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—81no—
240C42—24no—
241C42—24no—
242C42—24no—
243C42—24no—
244C42—24no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42—81no—
249C42C2×D412yes—
250C42C2×D412yes—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C4×D412yes—
258C23C22×D411yescharacteristic
259Q8—216no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—22no—
268Q8—41no—
269Q8—41no—
270Q8C4×D414yes—
271Q8C4×D414yes—
272C2×C4—216no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4C4×D412yes—
314C2×C4#12012yes—
315C2×C4#12012yes—
316C2×C4C23×C411yescharacteristic
317C2×C4C22×D411yescharacteristic
318C2×C4C22×D411yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C8—22no—
323C8—22no—
324C8—41no—
325C8—41no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22C2×C4×D411yescharacteristic
334C22C2×#12011yescharacteristic
335C22#52511yescharacteristic
336C22#57711yescharacteristic
337C22#57811yescharacteristic
338C22#57811yescharacteristic
339C22#57811yescharacteristic
340C4—22no—
341C4—22no—
342C4—24no—
343C4—24no—
344C4—22no—
345C4—42no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4C2×C4×D411yescharacteristic
354C4C2×C4×D411yescharacteristic
355C4#52511yescharacteristic
356C4#52511yescharacteristic
357C2—42no—
358C2C4×#12011yescharacteristic
359C2#294711yescharacteristic
360C2#295011yescharacteristic
361C2#301111yescharacteristic
362C2#301311yescharacteristic
363C2#312111yescharacteristic
364C2#312711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#441C2semidirect
4. C2×C4×Q16C2semidirect
6. #1878C2semidirect
7. #1878C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect
13. #1763C2semidirect
14. C4×#103C2semidirect
33. #451C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5192
C2×#4512
C2×#4412
C2×C4×Q162
#23232
#18782
#18732
#18652
#18632
#18542
#17872
#17632
C4×#1032
#17502