Finite Group Databasebeta

← #62545

Subgroups — #62545

Order 256.

860 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic
2#3509C211yesmaximal, characteristic, complemented
3#3377C214yesmaximal, complemented
4#3327C214yesmaximal, complemented
5#3315C214yesmaximal, complemented
6#3313C214yesmaximal, complemented
7#3231C211yesmaximal, characteristic, complemented
8#3230C211yesmaximal, characteristic, complemented
9#3228C211yesmaximal, characteristic, complemented
10#3210C211yesmaximal, characteristic, complemented
11#3210C211yesmaximal, characteristic, complemented
12#3209C211yesmaximal, characteristic, complemented
13#3185C211yesmaximal, characteristic, complemented
14#3182C211yesmaximal, characteristic, complemented
15#3170C211yesmaximal, characteristic, complemented
16#3141C211yesmaximal, characteristic, complemented
17#3139C211yesmaximal, characteristic, complemented
18#3131C211yesmaximal, characteristic, complemented
19#2965C211yesmaximal, characteristic, complemented
20#557—21no—
21#557C2212yes—
22#556—21no—
23#556C2212yes—
24#556C2212yes—
25#555—21no—
26#554—21no—
27#553—21no—
28#553—21no—
29#552—21no—
30#552—21no—
31#552C2212yes—
32#552C2212yes—
33#548C2212yes—
34#548C2212yes—
35#547C2212yes—
36#542—21no—
37#542—21no—
38#541—21no—
39#541—21no—
40#537—21no—
41#536—21no—
42#533C2211yescharacteristic
43#533C2211yescharacteristic
44#533C2211yescharacteristic
45#533C2211yescharacteristic
46#533C2211yescharacteristic
47#533C2211yescharacteristic
48#533C2211yescharacteristic
49#532C2211yescharacteristic
50#532C2211yescharacteristic
51#532C2211yescharacteristic
52C2×C4⋊Q8C2211yescharacteristic
53C2×C4⋊Q8C2211yescharacteristic
54#520—21no—
55#519—21no—
56C4×#41C2211yescharacteristic
57#500C2211yescharacteristic
58#500C2211yescharacteristic
59#498C2211yescharacteristic
60#494C2211yescharacteristic
61#494C2211yescharacteristic
62#492C2211yescharacteristic
63#491C2214yes—
64#489C2214yes—
65#488C2214yes—
66#487C2211yescharacteristic
67#487C2211yescharacteristic
68#487C2211yescharacteristic
69#487C2211yescharacteristic
70#487C2211yescharacteristic
71#487C2211yescharacteristic
72#487C2211yescharacteristic
73#486C2211yescharacteristic
74#486C2211yescharacteristic
75#486C2211yescharacteristic
76#485C2214yes—
77#484C2214yes—
78#482C2214yes—
79#479C2214yes—
80#478C2214yes—
81#477C2214yes—
82#470C2214yes—
83#468C2214yes—
84#467C2214yes—
85#464C2214yes—
86#464C2214yes—
87#462C2214yes—
88#452C2214yes—
89#452C2214yes—
90#451C2214yes—
91#446C2212yes—
92#441C2214yes—
93#440C2214yes—
94C4×Q16C2214yes—
95#436C2212yes—
96#435C2212yes—
97#435C2212yes—
98#431C2211yescharacteristic
99#431C2211yescharacteristic
100#431C2211yescharacteristic
101#419C2214yes—
102#417C2214yes—
103#416C2214yes—
104C4×M16C2211yescharacteristic
105C4×M16C2211yescharacteristic
106C2×C4×C8C2211yescharacteristic
107C2×#41—21no—
108C2×#41D411yescharacteristic
109C2×#41C2312yes—
110C2×#41C2312yes—
111C22×Q8—21no—
112C22×Q8C2312yes—
113C2×Q16C2314yes—
114C2×Q16C2314yes—
115C2×SD16C2314yes—
116C2×M16C2312yes—
117C2×M16C2312yes—
118C22×C8C2312yes—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C2311yescharacteristic
125C4⋊Q8C2311yescharacteristic
126C4⋊Q8C2312yes—
127C4⋊Q8C2311yescharacteristic
128C4⋊Q8C2311yescharacteristic
129C4⋊Q8C2311yescharacteristic
130C4⋊Q8C2312yes—
131C4⋊Q8C2311yescharacteristic
132C4⋊Q8C2311yescharacteristic
133#127C2311yescharacteristic
134#126—22no—
135#126—22no—
136#126—22no—
137#126—22no—
138#126—22no—
139#126—22no—
140#125—22no—
141#125—22no—
142#125—22no—
143#125—22no—
144#125—21no—
145#125—21no—
146#125—21no—
147#125—21no—
148#125—21no—
149#125C2312yes—
150#125C2312yes—
151#124—22no—
152#124—22no—
153#124—22no—
154#124—21no—
155#124C2312yes—
156#124C2312yes—
157#123—22no—
158#123—22no—
159#123—22no—
160#123—22no—
161C22⋊Q8—22no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8C2312yes—
174C22⋊Q8C2312yes—
175C22⋊Q8C2312yes—
176C22⋊Q8C2312yes—
177C22⋊Q8C2312yes—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121C2312yes—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8D411yescharacteristic
199C4×Q8C2312yes—
200C4×Q8C2312yes—
201C4×Q8C2312yes—
202C4×Q8C2312yes—
203C4×Q8C2312yes—
204C4×Q8C2311yescharacteristic
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4D411yescharacteristic
213C4×D4D411yescharacteristic
214C4×D4D411yescharacteristic
215C4×D4C2312yes—
216C4×D4C2311yescharacteristic
217C4×D4C2311yescharacteristic
218C4×D4C2311yescharacteristic
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117D411yescharacteristic
228#117D411yescharacteristic
229#117D411yescharacteristic
230#117C2312yes—
231#117C2312yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2312yes—
237C2×C42C2311yescharacteristic
238C2×C42C2311yescharacteristic
239C2×C42C2311yescharacteristic
240#107C2314yes—
241#107C2314yes—
242#106C2314yes—
243C4⋊C8C2312yes—
244C4⋊C8C2312yes—
245C4⋊C8C2312yes—
246#103C2314yes—
247#103C2314yes—
248#103C2314yes—
249#103C2314yes—
250#103C2314yes—
251#103C2314yes—
252#103C2314yes—
253#102C2314yes—
254#102C2314yes—
255#102C2314yes—
256C22⋊C8C2312yes—
257C22⋊C8C2312yes—
258C22⋊C8C2312yes—
259#97C2312yes—
260#97C2312yes—
261C4×C8C2311yescharacteristic
262C4×C8C2311yescharacteristic
263C4×C8C2312yes—
264C4×C8C2311yescharacteristic
265C4×C8C2311yescharacteristic
266#41—42no—
267#41—44no—
268#41—44no—
269#41—44no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8C2×D411yescharacteristic
279C2×Q8C2412yes—
280C2×Q8C2412yes—
281C2×Q8C2412yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4C2×D411yescharacteristic
288C2×D4C2×D411yescharacteristic
289C2×D4C2×D411yescharacteristic
290C2×D4C2412yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4C2×D411yescharacteristic
297C22×C4C2×D411yescharacteristic
298C22×C4C2×D411yescharacteristic
299C22×C4C2×D411yescharacteristic
300C22×C4C2×D411yescharacteristic
301C22×C4C2×D411yescharacteristic
302C22×C4C2411yescharacteristic
303C22×C4C2411yescharacteristic
304C22×C4C2411yescharacteristic
305Q16—44no—
306Q16—44no—
307SD16—44no—
308M16—42no—
309M16—42no—
310C2×C8—42no—
311C2×C8C2412yes—
312C2×C8C2412yes—
313C2×C8C2412yes—
314C2×C8C2412yes—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4C2×D411yescharacteristic
344C4⋊C4C2×D411yescharacteristic
345C4⋊C4C2×D411yescharacteristic
346C4⋊C4C2×D411yescharacteristic
347C4⋊C4C2×D411yescharacteristic
348C4⋊C4C2×D411yescharacteristic
349C4⋊C4C2412yes—
350C4⋊C4C2412yes—
351C4⋊C4C2412yes—
352C4⋊C4C2412yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4C2×D411yescharacteristic
367C22⋊C4C2×D411yescharacteristic
368C22⋊C4C2×D411yescharacteristic
369C22⋊C4C2×D411yescharacteristic
370C22⋊C4C2×D411yescharacteristic
371C22⋊C4C2×D411yescharacteristic
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42C2×D411yescharacteristic
380C42C2×D411yescharacteristic
381C42C2×D411yescharacteristic
382C42C2×D411yescharacteristic
383C42C2×D411yescharacteristic
384C42C2×D411yescharacteristic
385C42C2411yescharacteristic
386C42C2411yescharacteristic
387C42C2411yescharacteristic
388C42C2411yescharacteristic
389C23—22no—
390C23C22×D411yescharacteristic
391C23C22×D411yescharacteristic
392C23C22×D411yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—24no—
396Q8—41no—
397Q8—44no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—82no—
403D4—24no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—44no—
408D4—42no—
409D4—42no—
410D4—82no—
411D4—82no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4C22×D411yescharacteristic
434C2×C4C22×D411yescharacteristic
435C2×C4C22×D411yescharacteristic
436C2×C4C22×D411yescharacteristic
437C2×C4C22×D411yescharacteristic
438C2×C4C22×D411yescharacteristic
439C2×C4C22×D411yescharacteristic
440C2×C4C22×D411yescharacteristic
441C2×C4C22×D411yescharacteristic
442C2×C4C22×D411yescharacteristic
443C2×C4C22×D411yescharacteristic
444C2×C4C2511yescharacteristic
445C8—22no—
446C8—22no—
447C8—22no—
448C8—22no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22C23×D411yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4#57512yes—
479C4#57812yes—
480C4#58512yes—
481C2—41no—
482C2—41no—
483C2—41no—
484C2—82no—
485C2#350011yescharacteristic
486C2C2×#57511yescharacteristic
487C2C2×#57811yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3511C2
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. #3377C2semidirect
4. #3327C2semidirect
5. #3315C2semidirect
6. #3313C2semidirect
7. #3231C2semidirect
8. #3230C2semidirect
9. #3228C2semidirect
10. #3210C2semidirect
11. #3210C2semidirect
12. #3209C2semidirect
13. #3185C2semidirect
14. #3182C2semidirect
15. #3170C2semidirect
16. #3141C2semidirect
17. #3139C2semidirect
18. #3131C2semidirect
19. #2965C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#35092
#33772
#33272
#33152
#33132
#32312
#32302
#32282
#32102
#32092
#31852
#31822
#31702
#31412
#31392
#31312
#29652