Finite Group Databasebeta

← #62508

Subgroups — #62508

Order 256.

857 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3518C211yesmaximal, characteristic, complemented
2#3511C211yesmaximal, characteristic, complemented
3#3385C212yesmaximal, complemented
4#3384C212yesmaximal, complemented
5#3350C214yesmaximal, complemented
6#3347C214yesmaximal, complemented
7#3341C212yesmaximal, complemented
8#3340C212yesmaximal, complemented
9#3240C212yesmaximal, complemented
10#3238C211yesmaximal, characteristic, complemented
11#3235C211yesmaximal, characteristic, complemented
12#3235C212yesmaximal, complemented
13#3180C212yesmaximal, complemented
14#3171C211yesmaximal, characteristic, complemented
15#3165C211yesmaximal, characteristic, complemented
16#3162C212yesmaximal, complemented
17#2970C211yesmaximal, characteristic, complemented
18Q82C2211yescharacteristic
19#556—21no—
20#556—21no—
21#556—22no—
22#556C2211yescharacteristic
23#556C2212yes—
24#555C2212yes—
25#555C2211yescharacteristic
26#555C2212yes—
27#554C2212yes—
28#553C2214yes—
29D4×Q8C2211yescharacteristic
30#548—21no—
31#548—22no—
32#547—21no—
33#542—21no—
34#542—22no—
35#542—22no—
36#537—21no—
37#533C2212yes—
38#533C2212yes—
39#533C2211yescharacteristic
40#532C2211yescharacteristic
41#532C2212yes—
42C2×C4⋊Q8C2211yescharacteristic
43#529C2211yescharacteristic
44#529C2212yes—
45#520—21no—
46#519—21no—
47C4×#41C2211yescharacteristic
48#501C2214yes—
49#501C2214yes—
50#501C2211yescharacteristic
51#501C2212yes—
52#501C2212yes—
53#501C2212yes—
54#501C2212yes—
55#500C2212yes—
56#498C2212yes—
57#496C2211yescharacteristic
58#490C2211yescharacteristic
59#490C2212yes—
60#489C2211yescharacteristic
61#489C2212yes—
62#485C2214yes—
63#483C2212yes—
64#482C2214yes—
65#482C2214yes—
66#482C2214yes—
67#482C2214yes—
68#480C2212yes—
69#479C2214yes—
70#478C2214yes—
71#477C2212yes—
72#476C2212yes—
73#476C2212yes—
74#476C2214yes—
75#474C2212yes—
76#474C2214yes—
77#469C2212yes—
78#468C2212yes—
79#466C2214yes—
80#465C2214yes—
81#446C2212yes—
82#442C2214yes—
83#440C2214yes—
84C4×SD16C2212yes—
85C4×D8C2212yes—
86#435C2214yes—
87C8×D4C2212yes—
88#432C2211yescharacteristic
89#432C2212yes—
90#428C2214yes—
91#427C2214yes—
92#427C2214yes—
93C4×M16C2211yescharacteristic
94C2×#97C2212yes—
95C2×#41—21no—
96C2×#41—22no—
97C2×#41D411yescharacteristic
98C22×Q8—21no—
99C22×Q8—21no—
100C2×SD16C2312yes—
101C2×D8C2312yes—
102C2×M16C2312yes—
103C22×C8C2314yes—
104C4⋊Q8—21no—
105C4⋊Q8—22no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8C2312yes—
109C4⋊Q8C2312yes—
110C4⋊Q8C2311yescharacteristic
111C4⋊Q8C2311yescharacteristic
112C4⋊Q8C2312yes—
113C4⋊Q8C2312yes—
114C4⋊Q8C2311yescharacteristic
115#127C2311yescharacteristic
116#126—24no—
117#126—22no—
118#126—22no—
119#126—22no—
120#126—22no—
121#126—21no—
122#125C2311yescharacteristic
123#125C2312yes—
124#125C2314yes—
125#124—21no—
126#124—22no—
127#124—24no—
128#124—22no—
129#124—22no—
130#124—22no—
131#124—21no—
132#124—22no—
133#124C2311yescharacteristic
134#123—21no—
135#123—22no—
136#123—24no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—21no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—21no—
143C22⋊Q8—22no—
144C22⋊Q8—24no—
145C22⋊Q8—21no—
146C22⋊Q8—22no—
147C22⋊Q8C2312yes—
148C22⋊Q8C2311yescharacteristic
149#121—22no—
150#121—24no—
151#121—21no—
152#121—22no—
153#121—22no—
154#121—21no—
155#121C2312yes—
156#121C2311yescharacteristic
157#121C2312yes—
158#121C2314yes—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8D411yescharacteristic
169C4×Q8C2311yescharacteristic
170C4×Q8C2312yes—
171C4×Q8C2311yescharacteristic
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—21no—
177C4×D4D411yescharacteristic
178C4×D4D412yes—
179C4×D4C2311yescharacteristic
180C4×D4C2312yes—
181C4×D4C2314yes—
182C4×D4C2312yes—
183C4×D4C2312yes—
184C4×D4C2311yescharacteristic
185#117—21no—
186#117—22no—
187#117—22no—
188#117—21no—
189#117—22no—
190#117D411yescharacteristic
191#117D412yes—
192#117C2314yes—
193#117C2312yes—
194#117C2312yes—
195#117C2312yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C2312yes—
199C2×C42C2311yescharacteristic
200C2×C42C2312yes—
201#107C2312yes—
202#107C2314yes—
203#107C2314yes—
204#107C2314yes—
205#107C2314yes—
206#106C2312yes—
207#106C2314yes—
208#106C2314yes—
209#106C2314yes—
210#106C2314yes—
211C4⋊C8C2312yes—
212C4⋊C8C2314yes—
213#103C2312yes—
214#103C2314yes—
215#102C2312yes—
216#102C2312yes—
217#102C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2314yes—
222C22⋊C8C2312yes—
223#97C2311yescharacteristic
224#97C2312yes—
225#97C2314yes—
226#97C2311yescharacteristic
227C4×C8C2312yes—
228#41—42no—
229#41—44no—
230#41—84no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8C2×D411yescharacteristic
241C2×Q8C2411yescharacteristic
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C2×D411yescharacteristic
249C2×D4C2×D412yes—
250C2×D4C2411yescharacteristic
251C2×D4C2412yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D412yes—
259C22×C4C2×D411yescharacteristic
260C22×C4C2×D412yes—
261C22×C4C2411yescharacteristic
262C22×C4C2412yes—
263SD16—24no—
264D8—24no—
265M16—24no—
266C2×C8—28no—
267C2×C8C2412yes—
268C2×C8C2414yes—
269C2×C8C2412yes—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4C2×D411yescharacteristic
289C4⋊C4C2×D411yescharacteristic
290C4⋊C4C2×D412yes—
291C4⋊C4C2×D412yes—
292C4⋊C4C2411yescharacteristic
293C4⋊C4C2412yes—
294C4⋊C4C2411yescharacteristic
295C4⋊C4C2412yes—
296C4⋊C4C2414yes—
297C4⋊C4C2412yes—
298C22⋊C4—24no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—21no—
304C22⋊C4—22no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—24no—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—22no—
312C22⋊C4—21no—
313C22⋊C4—22no—
314C22⋊C4C2×D412yes—
315C22⋊C4C2×D412yes—
316C22⋊C4C2×D412yes—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42C2×D411yescharacteristic
324C42C2×D412yes—
325C42C2×D412yes—
326C42C2×D411yescharacteristic
327C42C2411yescharacteristic
328C42C2411yescharacteristic
329C42C2412yes—
330C23—21no—
331C23—22no—
332C23C22×D412yes—
333C23C22×D411yescharacteristic
334Q8—22no—
335Q8—22no—
336Q8—44no—
337Q8—44no—
338Q8—42no—
339Q8—42no—
340Q8—82no—
341Q8—81no—
342Q8—82no—
343D4—22no—
344D4—22no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—44no—
351D4—81no—
352D4—82no—
353D4—84no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4C22×D412yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D412yes—
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C22×D411yescharacteristic
378C2×C4C22×D412yes—
379C2×C4C22×D411yescharacteristic
380C2×C4C2511yescharacteristic
381C8—24no—
382C8—22no—
383C82−1+414yes—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—42no—
396C22C23×D411yescharacteristic
397C4—21no—
398C4—22no—
399C4—21no—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—44no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4#57512yes—
412C4C2×2−1+412yes—
413C2—41no—
414C2—42no—
415C2—82no—
416C2—81no—
417C2#349911yescharacteristic
418C2C2×#57511yescharacteristic
419C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3518C2semidirect
2. #3511C2semidirect
3. #3385C2semidirect
4. #3384C2semidirect
5. #3350C2semidirect
6. #3347C2semidirect
7. #3341C2semidirect
8. #3340C2semidirect
9. #3240C2semidirect
10. #3238C2semidirect
11. #3235C2semidirect
12. #3235C2semidirect
13. #3180C2semidirect
14. #3171C2semidirect
15. #3165C2semidirect
16. #3162C2semidirect
17. #2970C2semidirect

Maximal subgroups

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

subgroupindex
#35182
#35112
#33852
#33842
#33502
#33472
#33412
#33402
#32402
#32382
#32352
#31802
#31712
#31652
#31622
#29702