Finite Group Databasebeta

← #62134

Subgroups — #62134

Order 256.

879 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic, complemented
2#3482C211yesmaximal, characteristic, complemented
3#3384C212yesmaximal, complemented
4#3372C212yesmaximal, complemented
5#3350C214yesmaximal, complemented
6#3340C212yesmaximal, complemented
7#3307C214yesmaximal, complemented
8#3303C212yesmaximal, complemented
9#3254C211yesmaximal, characteristic, complemented
10#3246C212yesmaximal, complemented
11#3237C212yesmaximal, complemented
12#3229C211yesmaximal, characteristic, complemented
13#3115C212yesmaximal, complemented
14#3106C211yesmaximal, characteristic, complemented
15#3079C211yesmaximal, characteristic, complemented
16#3076C212yesmaximal, complemented
17#2990C211yesmaximal, characteristic, complemented
18#557C2211yescharacteristic
19#556—21no—
20#556C2211yescharacteristic
21#556C2212yes—
22#555C2212yes—
23#554C2212yes—
24#553C2214yes—
25#552—21no—
26#552—22no—
27D4×Q8C2211yescharacteristic
28#548—21no—
29#548—22no—
30#547—21no—
31#547C2211yescharacteristic
32#547C2212yes—
33#542—22no—
34#537—21no—
35#533C2211yescharacteristic
36#533C2212yes—
37#533C2212yes—
38#532C2211yescharacteristic
39#532C2212yes—
40C2×C4⋊Q8C2211yescharacteristic
41#529—21no—
42#529—22no—
43#520—21no—
44C4×#41C2212yes—
45C4×#41C2211yescharacteristic
46C4×#41C2211yescharacteristic
47C2×C4×Q8—21no—
48#501C2214yes—
49#498C2212yes—
50#491C2214yes—
51#487C2212yes—
52#485C2214yes—
53#485C2214yes—
54#483C2214yes—
55#482C2214yes—
56#482C2212yes—
57#479C2214yes—
58#478C2214yes—
59#477C2212yes—
60#477C2212yes—
61#477C2211yescharacteristic
62#476C2211yescharacteristic
63#476C2212yes—
64#476C2214yes—
65#475C2212yes—
66#475C2212yes—
67#475C2212yes—
68#474C2212yes—
69#474C2212yes—
70#468C2214yes—
71#465C2214yes—
72#465C2214yes—
73#461C2214yes—
74#461C2212yes—
75#460C2211yescharacteristic
76#460C2212yes—
77#459C2211yescharacteristic
78#449C2212yes—
79#449C2214yes—
80#446C2212yes—
81#442C2214yes—
82#440C2214yes—
83C4×SD16C2212yes—
84C4×D8C2212yes—
85#435C2214yes—
86C8×D4C2212yes—
87#432C2212yes—
88#431C2211yescharacteristic
89#428C2212yes—
90#427C2214yes—
91#423C2211yescharacteristic
92C2×C4⋊C8C2212yes—
93#417C2212yes—
94#416C2214yes—
95C2×#41—21no—
96C2×#41—22no—
97C2×#41D411yescharacteristic
98C2×#41C2312yes—
99C2×#41C2311yescharacteristic
100C22×Q8—21no—
101C22×Q8—21no—
102C2×SD16C2312yes—
103C2×SD16C2314yes—
104C2×D8C2312yes—
105C2×M16C2312yes—
106C22×C8C2314yes—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8C2311yescharacteristic
110C4⋊Q8C2312yes—
111C4⋊Q8C2312yes—
112C4⋊Q8C2312yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2312yes—
116#127C2311yescharacteristic
117#126—24no—
118#126—22no—
119#126—22no—
120#126—22no—
121#125—21no—
122#125—22no—
123#125—22no—
124#125—21no—
125#125C2314yes—
126#124—21no—
127#124—22no—
128#124—24no—
129#124—21no—
130#124—22no—
131#123—22no—
132#123—22no—
133#123—22no—
134#123—24no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—21no—
139C22⋊Q8—22no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—24no—
143C22⋊Q8C2312yes—
144C22⋊Q8C2312yes—
145C22⋊Q8C2312yes—
146#121—21no—
147#121—22no—
148#121—22no—
149#121—24no—
150#121—22no—
151#121—21no—
152#121C2312yes—
153#121C2314yes—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—22no—
157C4×Q8—21no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8D411yescharacteristic
162C4×Q8C2312yes—
163C4×Q8C2311yescharacteristic
164C4×Q8C2311yescharacteristic
165C4×Q8C2312yes—
166C4×Q8C2311yescharacteristic
167C4×D4—21no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—21no—
174C4×D4D411yescharacteristic
175C4×D4D412yes—
176C4×D4C2311yescharacteristic
177C4×D4C2311yescharacteristic
178C4×D4C2312yes—
179C4×D4C2314yes—
180C4×D4C2312yes—
181C4×D4C2312yes—
182C4×D4C2311yescharacteristic
183#117—21no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—21no—
188#117D411yescharacteristic
189#117D412yes—
190#117C2312yes—
191#117C2311yescharacteristic
192#117C2314yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2312yes—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42C2311yescharacteristic
201C2×C42C2312yes—
202#107C2314yes—
203#107C2314yes—
204#107C2312yes—
205#106C2312yes—
206#106C2314yes—
207#106C2314yes—
208#106C2314yes—
209C4⋊C8C2311yescharacteristic
210C4⋊C8C2312yes—
211C4⋊C8C2314yes—
212C4⋊C8C2312yes—
213C4⋊C8C2311yescharacteristic
214#103C2314yes—
215#103C2314yes—
216#103C2314yes—
217#103C2312yes—
218#103C2314yes—
219#102C2312yes—
220#102C2314yes—
221#102C2312yes—
222#102C2312yes—
223#102C2314yes—
224C22⋊C8C2312yes—
225C22⋊C8C2314yes—
226#97C2314yes—
227C4×C8C2312yes—
228#41—24no—
229#41—22no—
230#41—42no—
231#41—42no—
232#41—41no—
233#41—44no—
234#41—84no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—82no—
242C2×Q8C2×D411yescharacteristic
243C2×Q8C2412yes—
244C2×Q8C2411yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—24no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C2×D411yescharacteristic
252C2×D4C2×D412yes—
253C2×D4C2412yes—
254C2×D4C2411yescharacteristic
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4C2×D411yescharacteristic
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2411yescharacteristic
268C22×C4C2412yes—
269SD16—42no—
270SD16—44no—
271D8—24no—
272M16—42no—
273C2×C8—44no—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C2×C8C2414yes—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—24no—
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D411yescharacteristic
298C4⋊C4C2×D412yes—
299C4⋊C4C2×D412yes—
300C4⋊C4C2411yescharacteristic
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2414yes—
304C4⋊C4C2412yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—21no—
310C22⋊C4—24no—
311C22⋊C4—21no—
312C22⋊C4—24no—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—22no—
318C22⋊C4C2×D412yes—
319C22⋊C4C2×D412yes—
320C22⋊C4C2×D412yes—
321C42—21no—
322C42—22no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—21no—
327C42—24no—
328C42C2×D411yescharacteristic
329C42C2×D412yes—
330C42C2×D412yes—
331C42C2×D411yescharacteristic
332C42C2411yescharacteristic
333C42C2412yes—
334C42C2411yescharacteristic
335C23—21no—
336C23—22no—
337C23C22×D412yes—
338C23C22×D411yescharacteristic
339Q8—22no—
340Q8—22no—
341Q8—21no—
342Q8—21no—
343Q8—22no—
344Q8—44no—
345Q8—42no—
346Q8—42no—
347Q8—44no—
348Q8—82no—
349D4—22no—
350D4—22no—
351D4—21no—
352D4—22no—
353D4—42no—
354D4—42no—
355D4—44no—
356D4—42no—
357D4—44no—
358D4—42no—
359D4—84no—
360D42−1+412yes—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4C22×D412yes—
385C2×C4C22×D411yescharacteristic
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C2511yescharacteristic
393C8—22no—
394C8—22no—
395C8—24no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22C23×D411yescharacteristic
409C4—22no—
410C4—21no—
411C4—21no—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—41no—
421C4—44no—
422C4#57512yes—
423C4#57712yes—
424C4C2×2−1+411yescharacteristic
425C4#58511yescharacteristic
426C2—41no—
427C2—42no—
428C2—42no—
429C2—82no—
430C2#348911yescharacteristic
431C2C2×#57511yescharacteristic
432C2C2×#57711yescharacteristic
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. #3511C2semidirect
2. #3482C2semidirect
3. #3384C2semidirect
4. #3372C2semidirect
5. #3350C2semidirect
6. #3340C2semidirect
7. #3307C2semidirect
8. #3303C2semidirect
9. #3254C2semidirect
10. #3246C2semidirect
11. #3237C2semidirect
12. #3229C2semidirect
13. #3115C2semidirect
14. #3106C2semidirect
15. #3079C2semidirect
16. #3076C2semidirect
17. #2990C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#34822
#33842
#33722
#33502
#33402
#33072
#33032
#32542
#32462
#32372
#32292
#31152
#31062
#30792
#30762
#29902