Finite Group Databasebeta

← #62135

Subgroups — #62135

Order 256.

878 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509C211yesmaximal, characteristic, complemented
2#3483C211yesmaximal, characteristic
3#3387C212yesmaximal, complemented
4#3368C212yesmaximal, complemented
5#3348C214yesmaximal, complemented
6#3340C212yesmaximal, complemented
7#3317C214yesmaximal, complemented
8#3304C212yesmaximal, complemented
9#3254C211yesmaximal, characteristic, complemented
10#3246C212yesmaximal, complemented
11#3235C212yesmaximal, complemented
12#3227C211yesmaximal, characteristic, complemented
13#3116C212yesmaximal, complemented
14#3107C211yesmaximal, characteristic, complemented
15#3079C211yesmaximal, characteristic, complemented
16#3076C212yesmaximal, complemented
17#2990C211yesmaximal, characteristic, complemented
18Q82C2211yescharacteristic
19#557—21no—
20#557C2212yes—
21#556—21no—
22#556C2212yes—
23#555—21no—
24#555—22no—
25#555C2211yescharacteristic
26#555C2212yes—
27#552—22no—
28#552C2214yes—
29#550—21no—
30#550C2211yescharacteristic
31#550C2212yes—
32#550C2211yescharacteristic
33#541—22no—
34#536—21no—
35#533C2211yescharacteristic
36#533C2212yes—
37#533C2212yes—
38#532—22no—
39#532—21no—
40#532C2211yescharacteristic
41#532C2212yes—
42C2×C4⋊Q8C2211yescharacteristic
43#519—21no—
44C4×#41—21no—
45C4×#41C2212yes—
46C4×#41C2211yescharacteristic
47C4×#41C2211yescharacteristic
48#501C2214yes—
49#498C2212yes—
50#490C2214yes—
51#486C2212yes—
52#485C2214yes—
53#485C2212yes—
54#482C2214yes—
55#482C2214yes—
56#481C2214yes—
57#479C2214yes—
58#478C2214yes—
59#477C2212yes—
60#477C2212yes—
61#476C2212yes—
62#476C2212yes—
63#476C2212yes—
64#475C2211yescharacteristic
65#475C2212yes—
66#475C2214yes—
67#474C2212yes—
68#474C2212yes—
69#474C2211yescharacteristic
70#468C2214yes—
71#465C2214yes—
72#465C2214yes—
73#462C2211yescharacteristic
74#461C2211yescharacteristic
75#461C2212yes—
76#460C2214yes—
77#460C2212yes—
78#452C2212yes—
79#452C2214yes—
80#446C2212yes—
81#441C2214yes—
82#440C2214yes—
83C4×Q16C2212yes—
84C4×SD16C2212yes—
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—21no—
97C2×#41—22no—
98C2×#41D411yescharacteristic
99C2×#41C2312yes—
100C2×#41C2311yescharacteristic
101C22×Q8—21no—
102C2×Q16C2312yes—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×M16C2312yes—
106C22×C8C2314yes—
107C4⋊Q8—22no—
108C4⋊Q8—21no—
109C4⋊Q8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2312yes—
114C4⋊Q8C2312yes—
115C4⋊Q8C2312yes—
116C4⋊Q8C2311yescharacteristic
117C4⋊Q8C2311yescharacteristic
118#127—21no—
119#127—22no—
120#127C2311yescharacteristic
121#127C2312yes—
122#126—24no—
123#126—24no—
124#125—24no—
125#125—24no—
126#125—21no—
127#125—22no—
128#125C2314yes—
129#124—22no—
130#124—22no—
131#124—22no—
132#124—22no—
133#124—22no—
134#124—21no—
135#123—24no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—21no—
139C22⋊Q8—22no—
140C22⋊Q8C2312yes—
141C22⋊Q8C2314yes—
142#121—21no—
143#121—22no—
144#121—21no—
145#121—22no—
146#121—21no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—22no—
151#121C2312yes—
152#121C2312yes—
153#121C2312yes—
154C4×Q8—21no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—21no—
161C4×Q8D411yescharacteristic
162C4×Q8C2311yescharacteristic
163C4×Q8C2311yescharacteristic
164C4×Q8C2312yes—
165C4×Q8C2314yes—
166C4×Q8C2312yes—
167C4×Q8C2311yescharacteristic
168C4×D4—21no—
169C4×D4—22no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—21no—
176C4×D4D412yes—
177C4×D4D411yescharacteristic
178C4×D4C2312yes—
179C4×D4C2311yescharacteristic
180C4×D4C2311yescharacteristic
181C4×D4C2312yes—
182C4×D4C2312yes—
183C4×D4C2311yescharacteristic
184#117—21no—
185#117—22no—
186#117—21no—
187#117—22no—
188#117—24no—
189#117—22no—
190#117—21no—
191#117D411yescharacteristic
192#117D412yes—
193#117C2312yes—
194#117C2311yescharacteristic
195#117C2314yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2312yes—
198C2×C42—22no—
199C2×C42—21no—
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#103C2312yes—
215#103C2314yes—
216#103C2312yes—
217#103C2312yes—
218#103C2314yes—
219#102C2314yes—
220#102C2314yes—
221#102C2314yes—
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—82no—
235#41—84no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—44no—
242C2×Q8C2×D411yescharacteristic
243C2×Q8C2412yes—
244C2×Q8C2411yescharacteristic
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4C2×D411yescharacteristic
255C2×D4C2×D412yes—
256C2×D4C2412yes—
257C2×D4C2411yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2411yescharacteristic
270C22×C4C2412yes—
271Q16—24no—
272SD16—42no—
273SD16—44no—
274M16—42no—
275C2×C8—44no—
276C2×C8C2412yes—
277C2×C8C2412yes—
278C2×C8C2414yes—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D412yes—
300C4⋊C4C2414yes—
301C4⋊C4C2412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2412yes—
304C4⋊C4C2411yescharacteristic
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—24no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—21no—
315C22⋊C4—22no—
316C22⋊C4—21no—
317C22⋊C4C2×D412yes—
318C22⋊C4C2×D412yes—
319C22⋊C4C2×D412yes—
320C42—21no—
321C42—22no—
322C42—21no—
323C42—22no—
324C42—21no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42C2×D412yes—
329C42C2×D411yescharacteristic
330C42C2×D412yes—
331C42C2×D411yescharacteristic
332C42C2411yescharacteristic
333C42C2412yes—
334C42C2411yescharacteristic
335C23—22no—
336C23—21no—
337C23C22×D412yes—
338C23C22×D411yescharacteristic
339Q8—22no—
340Q8—21no—
341Q8—22no—
342Q8—22no—
343Q8—44no—
344Q8—44no—
345Q8—42no—
346Q8—82no—
347Q8—81no—
348Q82−1+412yes—
349D4—22no—
350D4—22no—
351D4—21no—
352D4—21no—
353D4—22no—
354D4—42no—
355D4—44no—
356D4—42no—
357D4—44no—
358D4—42no—
359D4—82no—
360D4—81no—
361D4—82no—
362D4—82no—
363D4—82no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4C22×D412yes—
387C2×C4C22×D412yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C2×C4C2511yescharacteristic
395C8—22no—
396C8—22no—
397C8—24no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—21no—
404C22—42no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22C23×D411yescharacteristic
411C4—22no—
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—42no—
419C4—41no—
420C4—44no—
421C4—42no—
422C4—41no—
423C4—42no—
424C4#57512yes—
425C4#57712yes—
426C4C2×2−1+411yescharacteristic
427C4#58511yescharacteristic
428C2—41no—
429C2—42no—
430C2—82no—
431C2—81no—
432C2#348911yescharacteristic
433C2C2×#57511yescharacteristic
434C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3483C2
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. #3509C2semidirect
3. #3387C2semidirect
4. #3368C2semidirect
5. #3348C2semidirect
6. #3340C2semidirect
7. #3317C2semidirect
8. #3304C2semidirect
9. #3254C2semidirect
10. #3246C2semidirect
11. #3235C2semidirect
12. #3227C2semidirect
13. #3116C2semidirect
14. #3107C2semidirect
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
#35092
#34832
#33872
#33682
#33482
#33402
#33172
#33042
#32542
#32462
#32352
#32272
#31162
#31072
#30792
#30762
#29902