Finite Group Databasebeta

← #36828

Subgroups — #36828

Order 256.

887 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2854C212yesmaximal, complemented
2#2807C212yesmaximal, complemented
3#2803C212yesmaximal, complemented
4#2704C212yesmaximal, complemented
5#2701C212yesmaximal, complemented
6#2694C212yesmaximal, complemented
7#2687C212yesmaximal, complemented
8#2682C212yesmaximal, complemented
9#2671C212yesmaximal, complemented
10#2651C212yesmaximal, complemented
11#2610C211yesmaximal, characteristic, complemented
12#2606C211yesmaximal, characteristic, complemented
13#2598C211yesmaximal, characteristic, complemented
14#2558C212yesmaximal, complemented
15#2540C211yesmaximal, characteristic, complemented
16#2476C212yesmaximal, complemented
17#2337C211yesmaximal, characteristic, complemented
18#2319C211yesmaximal, characteristic, complemented
19#2296C211yesmaximal, characteristic, complemented
20C2×#125C2212yes—
21C2×#124C2211yescharacteristic
22C2×#123C2212yes—
23C2×#123C2212yes—
24C2×#123C2212yes—
25C2×C22⋊Q8C2212yes—
26C2×#121C2212yes—
27C2×#121C2212yes—
28C2×#121C2212yes—
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#400C2211yescharacteristic
34#400C2212yes—
35#399C2211yescharacteristic
36#399C2212yes—
37#399C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#398C2212yes—
42#398C2212yes—
43#397C2212yes—
44#397C2212yes—
45#396C2212yes—
46#396C2212yes—
47#396C2212yes—
48#395C2211yescharacteristic
49#394C2211yescharacteristic
50#394C2211yescharacteristic
51#394C2212yes—
52#394C2212yes—
53#394C2212yes—
54#394C2212yes—
55#394C2212yes—
56#394C2212yes—
57#393C2211yescharacteristic
58#392C2211yescharacteristic
59#392C2211yescharacteristic
60#392C2212yes—
61#391C2212yes—
62#390C2212yes—
63#390C2212yes—
64#390C2212yes—
65#390C2212yes—
66#390C2212yes—
67#389C2212yes—
68#389C2212yes—
69#389C2212yes—
70#389C2212yes—
71#389C2212yes—
72#389C2212yes—
73#388C2212yes—
74#388C2212yes—
75#388C2212yes—
76#388C2212yes—
77#388C2212yes—
78#388C2212yes—
79#388C2212yes—
80#388C2212yes—
81#388C2212yes—
82#388C2212yes—
83#388C2212yes—
84#388C2212yes—
85#387C2212yes—
86#387C2212yes—
87#386C2212yes—
88#386C2212yes—
89#386C2212yes—
90#386C2212yes—
91#385C2212yes—
92#385C2212yes—
93#385C2212yes—
94#385C2212yes—
95#385C2212yes—
96#385C2212yes—
97#383C2211yescharacteristic
98#381C2212yes—
99#380C2212yes—
100#380C2212yes—
101C4×C4⋊C4C2211yescharacteristic
102C4×C4⋊C4C2211yescharacteristic
103C4×C4⋊C4C2211yescharacteristic
104C4×C4⋊C4C2211yescharacteristic
105C4×C22⋊C4C2212yes—
106#376C2211yescharacteristic
107C22×Q8C2311yescharacteristic
108C22×D4C2311yescharacteristic
109C22×D4C2312yes—
110C22×D4C2312yes—
111C22×D4C2312yes—
112C22×D4C2311yescharacteristic
113C22×D4C2312yes—
114C23×C4C2311yescharacteristic
115C23×C4C2312yes—
116C23×C4C2312yes—
117C23×C4C2312yes—
118#125—44no—
119#124—24no—
120#123—44no—
121#123—44no—
122#123—44no—
123C22⋊Q8—44no—
124#121—44no—
125#121—44no—
126#121—44no—
127C4×D4—24no—
128#117—24no—
129#117—42no—
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2312yes—
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2312yes—
145C2×C4⋊C4C2312yes—
146C2×C4⋊C4C2312yes—
147C2×C4⋊C4C2312yes—
148C2×C4⋊C4C2312yes—
149C2×C4⋊C4C2312yes—
150C2×C4⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2311yescharacteristic
153C2×C22⋊C4C2311yescharacteristic
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2312yes—
161C2×C22⋊C4C2312yes—
162C2×C22⋊C4C2312yes—
163C2×C22⋊C4C2312yes—
164C2×C22⋊C4C2312yes—
165C2×C22⋊C4C2312yes—
166C2×C22⋊C4C2312yes—
167C2×C22⋊C4C2312yes—
168C2×C22⋊C4C2312yes—
169C2×C22⋊C4C2312yes—
170C2×C22⋊C4C2312yes—
171C2×C22⋊C4C2312yes—
172C2×C22⋊C4C2312yes—
173C2×C22⋊C4C2312yes—
174C2×C22⋊C4C2312yes—
175C2×C22⋊C4C2312yes—
176C2×C42—42no—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2311yescharacteristic
183C2×C42C2311yescharacteristic
184C2×C42C2312yes—
185C2×C42C2311yescharacteristic
186#95C2311yescharacteristic
187#95C2311yescharacteristic
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2311yescharacteristic
199#95C2312yes—
200#95C2311yescharacteristic
201#95C2312yes—
202#95C2312yes—
203#95C2312yes—
204#95C2312yes—
205#95C2312yes—
206#95C2312yes—
207C24C2412yes—
208C24C2412yes—
209C24C2411yescharacteristic
210C2×Q8—22no—
211C2×Q8—42no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C2411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2412yes—
255C22×C4C2411yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C22⋊C4—24no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C42—22no—
303C42—41no—
304C42—41no—
305C42—42no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42—42no—
310C42#4114yes—
311C23—21no—
312C23—22no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23H(5,𝔽2)12yes—
330C23C2511yescharacteristic
331Q8—82no—
332D4—44no—
333D4—82no—
334D4—84no—
335D4—84no—
336D4—84no—
337D4—84no—
338C2×C4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—48no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4C2×#4112yes—
382C2×C4C2×#4112yes—
383C2×C4C2×#4112yes—
384C2×C4H(5,𝔽2)14yes—
385C2×C42−1+412yes—
386C22—22no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22C22×#4111yescharacteristic
406C22C2×H(5,𝔽2)11yescharacteristic
407C22C2×H(5,𝔽2)12yes—
408C22C2×2−1+411yescharacteristic
409C22#58512yes—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—41no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—42no—
426C2—42no—
427C2—82no—
428C2—82no—
429C2#345511yescharacteristic
430C2#349211yescharacteristic
431C2#350811yescharacteristic
432C2#353412yes—
433C2#354712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2854C2semidirect
2. #2807C2semidirect
3. #2803C2semidirect
4. #2704C2semidirect
5. #2701C2semidirect
6. #2694C2semidirect
7. #2687C2semidirect
8. #2682C2semidirect
9. #2671C2semidirect
10. #2651C2semidirect
11. #2610C2semidirect
12. #2606C2semidirect
13. #2598C2semidirect
14. #2558C2semidirect
15. #2540C2semidirect
16. #2476C2semidirect
17. #2337C2semidirect
18. #2319C2semidirect
19. #2296C2semidirect

Maximal subgroups

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

subgroupindex
#28542
#28072
#28032
#27042
#27012
#26942
#26872
#26822
#26712
#26512
#26102
#26062
#25982
#25582
#25402
#24762
#23372
#23192
#22962