Finite Group Databasebeta

← #60594

Subgroups — #60594

Order 256.

995 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509C211yesmaximal, characteristic, complemented
2C4×2−1+4C211yesmaximal, characteristic
3#3127C214yesmaximal, complemented
4#3120C214yesmaximal, complemented
5#3116C212yesmaximal, complemented
6#3113C212yesmaximal, complemented
7#3107C212yesmaximal, complemented
8#3104C212yesmaximal, complemented
9#3097C212yesmaximal, complemented
10#3078C211yesmaximal, characteristic, complemented
11#3076C211yesmaximal, characteristic, complemented
12#3039C211yesmaximal, characteristic, complemented
13C2×#462C211yesmaximal, characteristic, complemented
14#3011C212yesmaximal
15#2950C212yesmaximal, complemented
16#2947C212yesmaximal, complemented
17#2925C211yesmaximal, characteristic, complemented
18C2×2−1+4C2211yescharacteristic
19C2×#137C2212yes—
20C22×Q16C2212yes—
21#557—21no—
22#557—21no—
23#556—21no—
24#556—21no—
25#552—22no—
26#552—22no—
27#541C2214yes—
28#536C2212yes—
29#533C2212yes—
30#533C2212yes—
31#533C2211yescharacteristic
32C2×C4⋊Q8C2211yescharacteristic
33#520—22no—
34#520—21no—
35#520C2212yes—
36#520C2211yescharacteristic
37#520C2211yescharacteristic
38#519—21no—
39#519C2212yes—
40#519C2211yescharacteristic
41#519C2212yes—
42C4×#41—22no—
43C4×#41—21no—
44C4×#41C2211yescharacteristic
45C4×#41C2212yes—
46C4×#41C2211yescharacteristic
47C2×C4×Q8—21no—
48C2×C4×Q8C2211yescharacteristic
49C2×C4×Q8C2212yes—
50#485C2214yes—
51#485C2214yes—
52#484C2214yes—
53#483C2214yes—
54#479C2212yes—
55#479C2214yes—
56#478C2212yes—
57#477C2211yescharacteristic
58#477C2212yes—
59#476C2211yescharacteristic
60#475C2211yescharacteristic
61#475C2212yes—
62#474C2211yescharacteristic
63#462C2212yes—
64#462C2212yes—
65#462C2212yes—
66#462C2214yes—
67#462C2212yes—
68#461C2212yes—
69#461C2212yes—
70#452C2214yes—
71#452C2214yes—
72#451C2214yes—
73#451C2214yes—
74#441C2214yes—
75#441C2214yes—
76#441C2214yes—
77#440C2214yes—
78C4×Q16C2214yes—
79C4×Q16C2214yes—
80C4×Q16C2214yes—
81C4×SD16C2214yes—
82#433C2214yes—
83#432C2212yes—
84#431C2212yes—
85#428C2212yes—
86#427C2212yes—
87#423C2211yescharacteristic
88C2×C4⋊C8C2211yescharacteristic
89#419C2212yes—
90#419C2212yes—
91#416C2212yes—
92C2×#103C2212yes—
93#408C2212yes—
94#405C2212yes—
952−1+4—24no—
962−1+4—24no—
97C2×#41—22no—
98C2×#41—21no—
99C2×#41C2312yes—
100C2×#41C2311yescharacteristic
101C2×#41C2311yescharacteristic
102C22×Q8—21no—
103C22×Q8C2312yes—
104C22×Q8C2312yes—
105C22×Q8C2311yescharacteristic
106#137—28no—
107C2×Q16—28no—
108C2×Q16C2314yes—
109C2×Q16C2314yes—
110C2×Q16C2314yes—
111C2×SD16C2314yes—
112C2×M16C2312yes—
113C22×C8C2312yes—
114C4⋊Q8—22no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C2312yes—
118C4⋊Q8C2312yes—
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2311yescharacteristic
121#126—24no—
122#126—24no—
123#125—24no—
124#125—24no—
125#125—22no—
126#125—22no—
127#125C2312yes—
128#124—22no—
129#123—24no—
130C22⋊Q8—21no—
131C22⋊Q8—21no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8C2314yes—
135C22⋊Q8C2314yes—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—22no—
146C4×Q8—21no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8D411yescharacteristic
152C4×Q8D411yescharacteristic
153C4×Q8C2312yes—
154C4×Q8C2311yescharacteristic
155C4×Q8C2312yes—
156C4×Q8C2312yes—
157C4×Q8C2312yes—
158C4×Q8C2312yes—
159C4×Q8C2311yescharacteristic
160C4×Q8C2314yes—
161C4×Q8C2314yes—
162C4×D4—22no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—22no—
166C4×D4—21no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4D412yes—
172C4×D4D411yescharacteristic
173C4×D4D412yes—
174C4×D4D411yescharacteristic
175C4×D4C2311yescharacteristic
176C4×D4C2312yes—
177C4×D4C2311yescharacteristic
178#117—22no—
179#117—22no—
180#117—21no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—24no—
186#117—22no—
187#117—21no—
188#117C2311yescharacteristic
189#117C2312yes—
190#117C2312yes—
191#117C2311yescharacteristic
192#117C2312yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C2311yescharacteristic
200C2×C4⋊C4C2312yes—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42C2311yescharacteristic
206C2×C42C2312yes—
207#107C2312yes—
208#107C2312yes—
209#107C2314yes—
210#106C2312yes—
211#106C2312yes—
212#106C2314yes—
213C4⋊C8C2311yescharacteristic
214C4⋊C8C2311yescharacteristic
215C4⋊C8C2312yes—
216C4⋊C8C2312yes—
217C4⋊C8C2312yes—
218#103C2314yes—
219#103C2314yes—
220#103C2312yes—
221#103C2312yes—
222#103C2314yes—
223#103C2314yes—
224#103C2314yes—
225#103C2314yes—
226#102C2312yes—
227#102C2312yes—
228C22⋊C8C2314yes—
229C22⋊C8C2314yes—
230#97C2314yes—
231C4×C8C2314yes—
232#41—216no—
233#41—28no—
234#41—24no—
235#41—24no—
236#41—22no—
237#41—22no—
238#41—22no—
239#41#4114yes—
240C2×Q8—28no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8C2×D411yescharacteristic
252C2×Q8#4114yes—
253C2×Q8C2414yes—
254C2×Q8C2411yescharacteristic
255C2×Q8C2412yes—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4C2×D412yes—
259C2×D4C2×D411yescharacteristic
260C2×D4C2411yescharacteristic
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D412yes—
273C22×C4C2×D411yescharacteristic
274C22×C4C2×D412yes—
275C22×C4C2412yes—
276C22×C4C2411yescharacteristic
277Q16—28no—
278Q16—28no—
279Q16—28no—
280SD16—28no—
281M16—24no—
282C2×C8—24no—
283C2×C8C2412yes—
284C2×C8C2414yes—
285C2×C8C2412yes—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4C2×D411yescharacteristic
308C4⋊C4C2×D412yes—
309C4⋊C4C2×D411yescharacteristic
310C4⋊C4C2×D412yes—
311C4⋊C4C2412yes—
312C4⋊C4C2412yes—
313C4⋊C4C2412yes—
314C4⋊C4C2411yescharacteristic
315C4⋊C4C2411yescharacteristic
316C22⋊C4—22no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—21no—
322C22⋊C4—22no—
323C22⋊C4—21no—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C42—22no—
328C42—21no—
329C42—22no—
330C42—21no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—21no—
336C42—24no—
337C42—24no—
338C42—22no—
339C42C2×D412yes—
340C42C2×D411yescharacteristic
341C42C2×D412yes—
342C42C2×D411yescharacteristic
343C42C2412yes—
344C42C2411yescharacteristic
345C42C2411yescharacteristic
346C23—21no—
347C23C22×D412yes—
348C23C22×D411yescharacteristic
349Q8—28no—
350Q8—22no—
351Q8—24no—
352Q8—21no—
353Q8—22no—
354Q8—24no—
355Q8—24no—
356Q8—42no—
357Q8—44no—
358Q8—42no—
359Q8C2×#4112yes—
360Q8C2×#4114yes—
361D4—28no—
362D4—24no—
363D4—24no—
364D4—22no—
365D4—21no—
366D4C2×#4112yes—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4C22×D412yes—
392C2×C4C22×D412yes—
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D412yes—
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C2×C4C2×#4112yes—
400C2×C4C2×#4112yes—
401C2×C4C2511yescharacteristic
402C8—22no—
403C8—22no—
404C8—24no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22#52512yes—
414C22C23×D411yescharacteristic
415C4—21no—
416C4—22no—
417C4—21no—
418C4—24no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—21no—
423C4—42no—
424C4—41no—
425C4—42no—
426C4—44no—
427C4—41no—
428C4—42no—
429C4#52512yes—
430C4#57812yes—
431C4C22×#4111yescharacteristic
432C4C22×#4111yescharacteristic
433C2—22no—
434C2—42no—
435C2—42no—
436C2C2×#52511yescharacteristic
437C2C2×#57811yescharacteristic
438C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×2−1+4C22
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. #3127C2semidirect
4. #3120C2semidirect
5. #3116C2semidirect
6. #3113C2semidirect
7. #3107C2semidirect
8. #3104C2semidirect
9. #3097C2semidirect
10. #3078C2semidirect
11. #3076C2semidirect
12. #3039C2semidirect
13. C2×#462C2semidirect
15. #2950C2semidirect
16. #2947C2semidirect
17. #2925C2semidirect

Maximal subgroups

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

subgroupindex
#35092
C4×2−1+42
#31272
#31202
#31162
#31132
#31072
#31042
#30972
#30782
#30762
#30392
C2×#4622
#30112
#29502
#29472
#29252