Finite Group Databasebeta

← #33187

Subgroups — #33187

Order 256.

584 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#496C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4C2×#466C211yesmaximal, characteristic, complemented
5C2×#465C211yesmaximal, characteristic, complemented
6C2×#405C211yesmaximal, characteristic, complemented
7#2620C211yesmaximal, characteristic, complemented
8#2619C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1972C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1968C211yesmaximal, characteristic, complemented
13#1958C211yesmaximal, characteristic, complemented
14#1957C211yesmaximal, characteristic, complemented
15#1847C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#496—24no—
30#469—24no—
31#468—24no—
32#466—24no—
33#465—24no—
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46#405—24no—
47C2×#97C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50C2×C4×C8C2211yescharacteristic
51#396—22no—
52#394—22no—
53#394—22no—
54#392—22no—
55#391C2211yescharacteristic
56#390C2211yescharacteristic
57#390C2211yescharacteristic
58#390C2211yescharacteristic
59#384C2211yescharacteristic
60#382C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—22no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16D414yescomplemented
85C22×C8—22no—
86C22×C8—22no—
87C22×C8D414yescomplemented
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#127—22no—
93#127—22no—
94#124—22no—
95#124—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98#121—22no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—22no—
103#121—22no—
104#117—22no—
105#117—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4D412yes—
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4D412yes—
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#106—22no—
127#106—22no—
128#103—22no—
129#103—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#97—22no—
137#97D414yescomplemented
138C4×C8—22no—
139C4×C8—22no—
140#95—22no—
141#95—22no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—82no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4—82no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4C2411yescharacteristic
194SD16—44no—
195SD16—44no—
196D8—44no—
197D8—44no—
198M16—28no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—28no—
204C2×C8C2×D412yes—
205C2×C8C2×D412yes—
206C2×C8C2×D412yes—
207C2×C8C2×D412yes—
208C2×C8C2×D412yes—
209C2×C8C2×D412yes—
210C2×C8#4112yes—
211C2×C8#4112yes—
212C4⋊C4—24no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C42—21no—
233C42—21no—
234C42C2×D412yes—
235C42#4112yes—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23#12712yes—
256C23C22×D411yescharacteristic
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—88no—
274D4—84no—
275D4—84no—
276D4—84no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4#12114yes—
300C2×C4#12712yes—
301C2×C4C22×D411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C8—22no—
308C8—22no—
309C8#12114yes—
310C8#12114yes—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22C2×#12111yescharacteristic
332C22C2×#12711yescharacteristic
333C22#53211yescharacteristic
334C22#57612yes—
335C22#57712yes—
336C4—22no—
337C4—24no—
338C4—42no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4C2×#12111yescharacteristic
345C4C2×#12111yescharacteristic
346C4#53211yescharacteristic
347C4#53211yescharacteristic
348C2—42no—
349C2—161no—
350C2—161no—
351C2—161no—
352C2#285611yescharacteristic
353C2#306212yes—
354C2#315411yescharacteristic
355C2#315511yescharacteristic
356C2#317912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#496C2semidirect
2. C2×#469C2semidirect
3. C2×#468C2semidirect
4. C2×#466C2semidirect
5. C2×#465C2semidirect
6. C2×#405C2semidirect
7. #2620C2semidirect
8. #2619C2semidirect
9. #1973C2semidirect
10. #1972C2semidirect
11. #1970C2semidirect
12. #1968C2semidirect
13. #1958C2semidirect
14. #1957C2semidirect
15. #1847C2semidirect
84. C2×M16D4semidirect
87. C22×C8D4semidirect
137. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4962
C2×#4692
C2×#4682
C2×#4662
C2×#4652
C2×#4052
#26202
#26192
#19732
#19722
#19702
#19682
#19582
#19572
#18472