Finite Group Databasebeta

← #60721

Subgroups — #60721

Order 256.

949 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3512C211yesmaximal, characteristic, complemented
2#3511C211yesmaximal, characteristic, complemented
3#3181C212yesmaximal, complemented
4#3180C212yesmaximal, complemented
5#3179C214yesmaximal, complemented
6#3178C214yesmaximal, complemented
7#3172C212yesmaximal, complemented
8#3171C212yesmaximal, complemented
9C2×#501C211yesmaximal, characteristic, complemented
10#3150C211yesmaximal, characteristic, complemented
11#3142C211yesmaximal, characteristic, complemented
12#3141C211yesmaximal, characteristic, complemented
13#3061C212yesmaximal, complemented
14#3060C212yesmaximal, complemented
15#2952C212yesmaximal, complemented
16#2951C212yesmaximal, complemented
17#2875C211yesmaximal, characteristic, complemented
18C2×#135C2212yes—
19C2×#135C2212yes—
20C2×#131C2212yes—
21#556—21no—
22#555—21no—
23#554—21no—
24#553—22no—
25#553—22no—
26#553—22no—
27#553—21no—
28#553—21no—
29#552—21no—
30#552—21no—
31#548—22no—
32#547—21no—
33#542C2214yes—
34#541C2212yes—
35#540C2212yes—
36#537C2212yes—
37#537C2211yescharacteristic
38#537C2211yescharacteristic
39#533C2212yes—
40#532C2211yescharacteristic
41#532C2211yescharacteristic
42#532C2212yes—
43C2×C4⋊Q8C2211yescharacteristic
44#529C2212yes—
45#529C2212yes—
46C2×#125C2211yescharacteristic
47#520C2211yescharacteristic
48#520C2212yes—
49#520C2211yescharacteristic
50C4×#41C2211yescharacteristic
51#501C2212yes—
52#501C2214yes—
53#501C2212yes—
54#497C2211yescharacteristic
55#497C2212yes—
56#497C2211yescharacteristic
57#496C2211yescharacteristic
58#496C2212yes—
59#496C2211yescharacteristic
60#491C2212yes—
61#490C2212yes—
62#489C2212yes—
63#489C2212yes—
64#488C2214yes—
65#487C2212yes—
66#486C2212yes—
67#470C2214yes—
68#469C2214yes—
69#468C2214yes—
70#468C2214yes—
71#467C2214yes—
72#466C2214yes—
73#465C2214yes—
74#465C2214yes—
75#442C2214yes—
76#441C2214yes—
77#440C2214yes—
78#440C2214yes—
79C4×Q16C2214yes—
80C4×SD16C2214yes—
81C4×SD16C2214yes—
82C4×D8C2214yes—
83C2×#107C2212yes—
84C2×#106C2212yes—
85#417C2212yes—
86#417C2212yes—
87#417C2212yes—
88#417C2212yes—
89#405C2214yes—
90#405C2212yes—
91C4×M16C2211yescharacteristic
92C4×M16C2212yes—
93C2×#97C2211yescharacteristic
94C2×#97C2212yes—
95C2×#41—22no—
96C2×#41C2311yescharacteristic
97C2×#41C2312yes—
98C2×#41C2311yescharacteristic
99C2×#41C2311yescharacteristic
100C22×Q8—21no—
101#135—28no—
102#135—28no—
103C2×Q16C2312yes—
104C2×Q16C2312yes—
105C2×SD16C2312yes—
106C2×SD16C2312yes—
107C2×SD16C2312yes—
108C2×SD16C2312yes—
109C2×D8C2312yes—
110C2×D8C2312yes—
111#131—28no—
112C2×M16C2312yes—
113C2×M16C2314yes—
114C22×C8C2314yes—
115C22×C8C2312yes—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2312yes—
121C4⋊Q8C2312yes—
122#127C2311yescharacteristic
123#127C2311yescharacteristic
124#126—24no—
125#126—22no—
126#126—22no—
127#126—22no—
128#126—22no—
129#125—22no—
130#125—22no—
131#125—21no—
132#125—21no—
133#125—22no—
134#125C2312yes—
135#125C2312yes—
136#124—24no—
137#124—22no—
138#124C2311yescharacteristic
139#124C2311yescharacteristic
140#124C2312yes—
141#123—22no—
142#123—22no—
143#123—22no—
144#123—22no—
145#123—24no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—24no—
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153C22⋊Q8C2314yes—
154#121—22no—
155#121—24no—
156#121—21no—
157#121—21no—
158#121—22no—
159#121—22no—
160#121—21no—
161#121—21no—
162#121—21no—
163#121—21no—
164#121—22no—
165#121—22no—
166#121—22no—
167#121C2312yes—
168#121C2312yes—
169#121C2314yes—
170C4×Q8—22no—
171C4×Q8D411yescharacteristic
172C4×Q8D411yescharacteristic
173C4×Q8C2312yes—
174C4×Q8C2312yes—
175C4×Q8C2314yes—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4D411yescharacteristic
187C4×D4D412yes—
188C4×D4D412yes—
189C4×D4D411yescharacteristic
190C4×D4C2312yes—
191C4×D4C2312yes—
192C4×D4C2314yes—
193#117—22no—
194#117—21no—
195#117—21no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117C2311yescharacteristic
200#117C2312yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2311yescharacteristic
208C2×C4⋊C4C2311yescharacteristic
209C2×C4⋊C4C2312yes—
210C2×C42C2311yescharacteristic
211C2×C42C2312yes—
212#107C2314yes—
213#107C2314yes—
214#106C2314yes—
215#106C2314yes—
216#103C2314yes—
217#103C2314yes—
218#103C2314yes—
219#103C2314yes—
220#102C2314yes—
221#102C2314yes—
222#102C2314yes—
223#102C2314yes—
224#97C2311yescharacteristic
225#97C2312yes—
226#97C2311yescharacteristic
227#97C2314yes—
228#97C2312yes—
229C4×C8C2314yes—
230C4×C8C2312yes—
231#41—24no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—24no—
236#41—84no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—82no—
240C2×Q8C2×D411yescharacteristic
241C2×Q8C2412yes—
242C2×Q8C2411yescharacteristic
243C2×Q8C2411yescharacteristic
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4C2×D412yes—
255C2×D4C2×D411yescharacteristic
256C2×D4C2411yescharacteristic
257C2×D4C2412yes—
258C2×D4C2411yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4C2×D411yescharacteristic
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D412yes—
271C22×C4C2×D412yes—
272C22×C4C2412yes—
273C22×C4C2411yescharacteristic
274Q16—24no—
275Q16—24no—
276SD16—24no—
277SD16—24no—
278SD16—24no—
279SD16—24no—
280D8—24no—
281D8—24no—
282M16—24no—
283M16—28no—
284C2×C8—28no—
285C2×C8#4118yescomplemented
286C2×C8C2414yes—
287C2×C8C2412yes—
288C2×C8C2412yes—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4C2×D411yescharacteristic
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D412yes—
311C4⋊C4C2×D412yes—
312C4⋊C4C2412yes—
313C4⋊C4C2414yes—
314C4⋊C4C2412yes—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—24no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4C2×D412yes—
331C22⋊C4C2×D412yes—
332C22⋊C4C2×D412yes—
333C42—22no—
334C42—22no—
335C42—24no—
336C42C2×D412yes—
337C42C2×D412yes—
338C42C2×D411yescharacteristic
339C42C2×D411yescharacteristic
340C42C2412yes—
341C42C2411yescharacteristic
342C42C2411yescharacteristic
343C23—22no—
344C23—21no—
345C23—21no—
346C23C22×D412yes—
347C23C22×D411yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—22no—
353Q8—44no—
354Q8—82no—
355D4—24no—
356D4—22no—
357D4—22no—
358D4—22no—
359D4—22no—
360D4—22no—
361D4—42no—
362D4—42no—
363D4—44no—
364D4—42no—
365D4—42no—
366D4—81no—
367D4—81no—
368D4—82no—
369D4—82no—
370D4—84no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4C22×D412yes—
392C2×C4C22×D412yes—
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D412yes—
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C2×C4C2×#4114yes—
400C2×C4C2511yescharacteristic
401C8—24no—
402C8C2×#4114yes—
403C8C2×#4114yes—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22#53212yes—
419C22C23×D411yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—22no—
425C4—21no—
426C4—22no—
427C4—42no—
428C4—44no—
429C4—42no—
430C4—41no—
431C4—41no—
432C4—42no—
433C4#53212yes—
434C4C22×#4112yes—
435C2—22no—
436C2—42no—
437C2—82no—
438C2—81no—
439C2—81no—
440C2C2×#53211yescharacteristic
441C2#358511yescharacteristic
442C2#358611yescharacteristic
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. #3512C2semidirect
2. #3511C2semidirect
3. #3181C2semidirect
4. #3180C2semidirect
5. #3179C2semidirect
6. #3178C2semidirect
7. #3172C2semidirect
8. #3171C2semidirect
9. C2×#501C2semidirect
10. #3150C2semidirect
11. #3142C2semidirect
12. #3141C2semidirect
13. #3061C2semidirect
14. #3060C2semidirect
15. #2952C2semidirect
16. #2951C2semidirect
17. #2875C2semidirect
285. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
#35122
#35112
#31812
#31802
#31792
#31782
#31722
#31712
C2×#5012
#31502
#31422
#31412
#30612
#30602
#29522
#29512
#28752