Finite Group Databasebeta

← #62511

Subgroups — #62511

Order 256.

829 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515C211yesmaximal, characteristic
2#3509C211yesmaximal, characteristic, complemented
3#3383C212yesmaximal, complemented
4#3381C212yesmaximal, complemented
5#3356C212yesmaximal, complemented
6#3353C212yesmaximal, complemented
7#3352C214yesmaximal, complemented
8#3351C214yesmaximal, complemented
9#3240C212yesmaximal, complemented
10#3239C211yesmaximal, characteristic, complemented
11#3237C211yesmaximal, characteristic, complemented
12#3236C212yesmaximal, complemented
13#3172C211yesmaximal, characteristic, complemented
14#3172C212yesmaximal, complemented
15#3166C212yesmaximal, complemented
16#3165C211yesmaximal, characteristic, complemented
17#2971C211yesmaximal, characteristic, complemented
18#557—21no—
19#557C2212yes—
20#557C2211yescharacteristic
21#556—21no—
22#556C2212yes—
23#554C2211yescharacteristic
24#554C2211yescharacteristic
25#553C2212yes—
26#552—22no—
27#552—22no—
28#552—21no—
29#552C2214yes—
30#552C2212yes—
31#547—21no—
32#547C2211yescharacteristic
33#541—21no—
34#541—22no—
35#539—22no—
36#536—21no—
37#533C2212yes—
38#533C2211yescharacteristic
39#533C2211yescharacteristic
40#533C2212yes—
41C2×C4⋊Q8C2211yescharacteristic
42#529C2211yescharacteristic
43#529C2212yes—
44C2×#125C2212yes—
45#520—21no—
46#519—21no—
47C4×#41C2211yescharacteristic
48#501C2212yes—
49#501C2212yes—
50#501C2211yescharacteristic
51#501C2212yes—
52#501C2212yes—
53#500C2212yes—
54#499C2214yes—
55#499C2214yes—
56#498C2212yes—
57#497C2211yescharacteristic
58#491C2211yescharacteristic
59#491C2212yes—
60#489C2212yes—
61#489C2211yescharacteristic
62#485C2214yes—
63#485C2214yes—
64#484C2212yes—
65#484C2214yes—
66#483C2214yes—
67#482C2214yes—
68#481C2212yes—
69#479C2214yes—
70#479C2214yes—
71#479C2214yes—
72#478C2214yes—
73#477C2212yes—
74#475C2212yes—
75#475C2212yes—
76#474C2212yes—
77#470C2212yes—
78#470C2214yes—
79#468C2212yes—
80#468C2214yes—
81C8×Q8C2212yes—
82C4×Q16C2214yes—
83C4×Q16C2212yes—
84C4×SD16C2214yes—
85C4×SD16C2212yes—
86#436C2214yes—
87#436C2212yes—
88#432C2212yes—
89#432C2211yescharacteristic
90#428C2214yes—
91#428C2214yes—
92#428C2214yes—
93C4×M16C2211yescharacteristic
94C4×M16C2212yes—
95C2×#41—21no—
96C2×#41D411yescharacteristic
97C22×Q8—21no—
98C2×Q16C2312yes—
99C2×SD16C2312yes—
100C2×M16C2312yes—
101C2×M16C2314yes—
102C4⋊Q8—22no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8C2311yescharacteristic
107C4⋊Q8C2312yes—
108C4⋊Q8C2312yes—
109C4⋊Q8C2311yescharacteristic
110C4⋊Q8C2311yescharacteristic
111C4⋊Q8C2311yescharacteristic
112#126—22no—
113#126—22no—
114#126—24no—
115#126—24no—
116#126—22no—
117#126—21no—
118#125—22no—
119#125—22no—
120#125—21no—
121#125—24no—
122#125—24no—
123#125—22no—
124#125—22no—
125#125—21no—
126#125—22no—
127#125C2311yescharacteristic
128#125C2312yes—
129#125C2312yes—
130#125C2312yes—
131#125C2314yes—
132#124—22no—
133#124C2311yescharacteristic
134#123—24no—
135#123—22no—
136#123—21no—
137#123—22no—
138#123—22no—
139#123—22no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—21no—
149C22⋊Q8—22no—
150C22⋊Q8C2311yescharacteristic
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153C22⋊Q8C2314yes—
154#121—22no—
155#121C2312yes—
156#121C2311yescharacteristic
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—22no—
165C4×Q8—21no—
166C4×Q8D411yescharacteristic
167C4×Q8C2312yes—
168C4×Q8C2311yescharacteristic
169C4×Q8C2314yes—
170C4×Q8C2312yes—
171C4×Q8C2311yescharacteristic
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—21no—
175C4×D4—22no—
176C4×D4D411yescharacteristic
177C4×D4D412yes—
178C4×D4C2311yescharacteristic
179C4×D4C2312yes—
180C4×D4C2311yescharacteristic
181C4×D4C2312yes—
182#117—21no—
183#117—22no—
184#117—24no—
185#117—22no—
186#117—21no—
187#117—22no—
188#117D411yescharacteristic
189#117D412yes—
190#117C2312yes—
191#117C2314yes—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2312yes—
197C2×C4⋊C4C2312yes—
198C2×C4⋊C4C2312yes—
199C2×C42C2312yes—
200C2×C42C2311yescharacteristic
201#107C2312yes—
202#107C2314yes—
203#107C2314yes—
204#107C2314yes—
205#107C2314yes—
206#106C2314yes—
207#106C2312yes—
208#106C2314yes—
209#106C2314yes—
210#106C2314yes—
211C4⋊C8C2312yes—
212C4⋊C8C2314yes—
213#103C2314yes—
214#103C2312yes—
215#103C2314yes—
216#103C2312yes—
217#103C2314yes—
218#103C2312yes—
219#102C2314yes—
220#102C2312yes—
221C22⋊C8C2312yes—
222C22⋊C8C2314yes—
223#97C2311yescharacteristic
224#97C2311yescharacteristic
225#97C2312yes—
226C4×C8C2312yes—
227C4×C8C2314yes—
228#41—42no—
229#41—44no—
230C2×Q8—21no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—44no—
236C2×Q8C2×D411yescharacteristic
237C2×Q8C2411yescharacteristic
238C2×Q8C2412yes—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C2×D411yescharacteristic
242C2×D4C2×D412yes—
243C2×D4C2411yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D412yes—
251C22×C4C2×D412yes—
252C22×C4C2×D411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255Q16—24no—
256SD16—24no—
257M16—24no—
258M16—28no—
259C2×C8C2412yes—
260C2×C8C2414yes—
261C2×C8C2412yes—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4C2×D411yescharacteristic
291C4⋊C4C2×D412yes—
292C4⋊C4C2×D411yescharacteristic
293C4⋊C4C2×D412yes—
294C4⋊C4C2414yes—
295C4⋊C4C2412yes—
296C4⋊C4C2411yescharacteristic
297C4⋊C4C2412yes—
298C4⋊C4C2411yescharacteristic
299C4⋊C4C2412yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—24no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—21no—
312C22⋊C4C2×D412yes—
313C22⋊C4C2×D412yes—
314C22⋊C4C2×D412yes—
315C42—22no—
316C42—24no—
317C42—22no—
318C42—24no—
319C42—21no—
320C42—22no—
321C42C2×D412yes—
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2×D412yes—
325C42C2412yes—
326C42C2411yescharacteristic
327C42C2411yescharacteristic
328C23—21no—
329C23C22×D412yes—
330C23C22×D411yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—22no—
335Q8—42no—
336Q8—44no—
337Q8—44no—
338Q8—42no—
339Q8—81no—
340D4—22no—
341D4—22no—
342D4—42no—
343D4—44no—
344D4—81no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C22×D412yes—
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D412yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C22×D412yes—
371C2×C4C22×D411yescharacteristic
372C2×C4C2511yescharacteristic
373C8—22no—
374C8—24no—
375C82−1+414yes—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22C23×D411yescharacteristic
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—42no—
400C4—44no—
401C4#57512yes—
402C4C2×2−1+412yes—
403C2—42no—
404C2—41no—
405C2—81no—
406C2#349911yescharacteristic
407C2C2×#57511yescharacteristic
408C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3515C2
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
2. #3509C2semidirect
3. #3383C2semidirect
4. #3381C2semidirect
5. #3356C2semidirect
6. #3353C2semidirect
7. #3352C2semidirect
8. #3351C2semidirect
9. #3240C2semidirect
10. #3239C2semidirect
11. #3237C2semidirect
12. #3236C2semidirect
13. #3172C2semidirect
14. #3172C2semidirect
15. #3166C2semidirect
16. #3165C2semidirect
17. #2971C2semidirect

Maximal subgroups

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

subgroupindex
#35152
#35092
#33832
#33812
#33562
#33532
#33522
#33512
#32402
#32392
#32372
#32362
#31722
#31662
#31652
#29712