Finite Group Databasebeta

← #62387

Subgroups — #62387

Order 256.

856 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3557C211yesmaximal, characteristic, complemented
2#3548C211yesmaximal, characteristic, complemented
3#3399C212yesmaximal, complemented
4#3397C212yesmaximal, complemented
5#3389C212yesmaximal, complemented
6#3388C212yesmaximal, complemented
7#3376C212yesmaximal, complemented
8#3370C212yesmaximal, complemented
9#3368C212yesmaximal, complemented
10#3366C212yesmaximal, complemented
11#3260C211yesmaximal, characteristic, complemented
12#3259C211yesmaximal, characteristic, complemented
13#3257C212yesmaximal, complemented
14#3251C212yesmaximal, complemented
15#3180C212yesmaximal, complemented
16#3179C212yesmaximal, complemented
17#3176C211yesmaximal, characteristic, complemented
18#3174C211yesmaximal, characteristic, complemented
19#2961C211yesmaximal, characteristic, complemented
20#563—22no—
21#562—22no—
22#559C2212yes—
23#557C2212yes—
24#556C2212yes—
25#554C2212yes—
26#554C2212yes—
27#554C2212yes—
28#552—22no—
29#550C2212yes—
30#550C2212yes—
31#547—22no—
32#542—22no—
33#542C2212yes—
34#541C2211yescharacteristic
35#540C2211yescharacteristic
36#540C2211yescharacteristic
37#540C2211yescharacteristic
38#540C2212yes—
39#539—22no—
40#537—22no—
41#537—22no—
42#533C2211yescharacteristic
43#532C2212yes—
44#529C2212yes—
45#501C2212yes—
46#501C2212yes—
47#499C2211yescharacteristic
48#499C2211yescharacteristic
49#496C2212yes—
50#496C2212yes—
51#493C2211yescharacteristic
52#492C2211yescharacteristic
53#491C2211yescharacteristic
54#490C2211yescharacteristic
55#490C2212yes—
56#490C2211yescharacteristic
57#490C2211yescharacteristic
58#489C2212yes—
59#488C2212yes—
60#486C2212yes—
61#479C2212yes—
62#479C2212yes—
63#478C2212yes—
64#478C2212yes—
65#478C2212yes—
66#478C2212yes—
67#478C2212yes—
68#478C2212yes—
69#476C2212yes—
70#475C2212yes—
71#474C2212yes—
72#474C2212yes—
73#469C2212yes—
74#469C2212yes—
75#468C2212yes—
76#468C2212yes—
77#466C2212yes—
78#466C2212yes—
79#465C2212yes—
80#465C2212yes—
81#463C2212yes—
82#463C2212yes—
83#463C2212yes—
84#463C2212yes—
85#461C2212yes—
86#460C2212yes—
87#460C2212yes—
88#460C2212yes—
89#459C2212yes—
90#459C2212yes—
91#459C2212yes—
92#459C2212yes—
93#446C2212yes—
94#446C2212yes—
95#446C2212yes—
96#446C2212yes—
97#442C2212yes—
98#442C2212yes—
99#442C2212yes—
100#442C2212yes—
101#440C2212yes—
102#440C2212yes—
103#440C2212yes—
104#440C2212yes—
105C4×SD16C2212yes—
106C4×SD16C2212yes—
107C4×D8C2212yes—
108C4×D8C2212yes—
109#424C2211yescharacteristic
110#424C2211yescharacteristic
111#423C2211yescharacteristic
112#405C2211yescharacteristic
113#405C2211yescharacteristic
114C2×#97C2211yescharacteristic
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×SD16C2312yes—
119C2×SD16C2312yes—
120C2×SD16C2312yes—
121C2×SD16C2312yes—
122C2×D8C2312yes—
123C2×D8C2312yes—
124C2×D8C2312yes—
125C2×D8C2312yes—
126C2×M16C2311yescharacteristic
127C2×M16C2311yescharacteristic
128C22×C8C2311yescharacteristic
129C22×C8C2311yescharacteristic
130C4⋊Q8—22no—
131C4⋊Q8D412yes—
132C4⋊Q8C2311yescharacteristic
133C4⋊Q8C2312yes—
134C4⋊Q8C2312yes—
135#127C2312yes—
136#127C2311yescharacteristic
137#127C2312yes—
138#127C2311yescharacteristic
139#127C2311yescharacteristic
140#126—22no—
141#126—22no—
142#126—22no—
143#126—22no—
144#126—22no—
145#125—22no—
146#125—22no—
147#125—22no—
148#125D412yes—
149#125C2312yes—
150#125C2311yescharacteristic
151#125C2312yes—
152#125C2311yescharacteristic
153#125C2311yescharacteristic
154#125C2311yescharacteristic
155#124—22no—
156#124—22no—
157#124—22no—
158#124C2312yes—
159#124C2312yes—
160#123—22no—
161#123—22no—
162#123—22no—
163#123—22no—
164#123—22no—
165#123—22no—
166#123—22no—
167#123—22no—
168#123—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8D412yes—
181C22⋊Q8D412yes—
182C22⋊Q8C2312yes—
183#121—22no—
184#121—22no—
185#121—22no—
186#121—22no—
187#121—22no—
188#121—22no—
189#121—22no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—22no—
194#121—22no—
195#121C2312yes—
196#121C2312yes—
197#121C2312yes—
198C4×Q8—22no—
199C4×Q8C2312yes—
200C4×Q8C2312yes—
201C4×Q8C2312yes—
202C4×Q8C2312yes—
203C4×Q8C2312yes—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4C2312yes—
208C4×D4C2312yes—
209C4×D4C2312yes—
210C4×D4C2312yes—
211C4×D4C2312yes—
212C4×D4C2312yes—
213C4×D4C2312yes—
214C4×D4C2312yes—
215C4×D4C2312yes—
216#117—22no—
217#117—22no—
218#117C2311yescharacteristic
219#117C2311yescharacteristic
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C42C2311yescharacteristic
224#107C2312yes—
225#107C2312yes—
226#107C2312yes—
227#107C2312yes—
228#106C2312yes—
229#106C2312yes—
230#106C2312yes—
231#106C2312yes—
232C4⋊C8C2312yes—
233C4⋊C8C2312yes—
234C4⋊C8C2312yes—
235C4⋊C8C2312yes—
236C4⋊C8C2312yes—
237C4⋊C8C2312yes—
238#103C2312yes—
239#103C2312yes—
240#103C2312yes—
241#103C2312yes—
242#102C2312yes—
243#102C2312yes—
244#102C2312yes—
245#102C2312yes—
246#102C2312yes—
247#102C2312yes—
248#102C2312yes—
249#102C2312yes—
250#102C2312yes—
251#102C2312yes—
252#102C2312yes—
253#102C2312yes—
254#97C2312yes—
255#97C2312yes—
256#97C2311yescharacteristic
257#97C2312yes—
258#97C2311yescharacteristic
259C4×C8C2311yescharacteristic
260C4×C8C2312yes—
261C4×C8C2311yescharacteristic
262#41—84no—
263#41—84no—
264#41—84no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8C2×D412yes—
268C2×Q8C2×D412yes—
269C2×Q8C2412yes—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4C2412yes—
281C2×D4C2412yes—
282C2×D4C2412yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4C2×D412yes—
293C22×C4C2411yescharacteristic
294SD16—24no—
295SD16—42no—
296SD16—42no—
297SD16—42no—
298D8—24no—
299D8—42no—
300D8—42no—
301D8—42no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8C2411yescharacteristic
307C2×C8C2411yescharacteristic
308C2×C8C2412yes—
309C2×C8C2411yescharacteristic
310C2×C8C2412yes—
311C2×C8C2411yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4C2×D412yes—
328C4⋊C4C2×D412yes—
329C4⋊C4C2×D412yes—
330C4⋊C4C2×D412yes—
331C4⋊C4C2×D412yes—
332C4⋊C4C2×D412yes—
333C4⋊C4C2×D412yes—
334C4⋊C4C2×D412yes—
335C4⋊C4C2412yes—
336C4⋊C4C2412yes—
337C4⋊C4C2412yes—
338C4⋊C4C2412yes—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4C2×D412yes—
357C22⋊C4C2×D411yescharacteristic
358C22⋊C4C2×D411yescharacteristic
359C42—22no—
360C42—22no—
361C42—22no—
362C42C2×D411yescharacteristic
363C42C2×D411yescharacteristic
364C42C2411yescharacteristic
365C42C2412yes—
366C42C2411yescharacteristic
367C42C2412yes—
368C23—22no—
369C23—22no—
370C23—22no—
371C23C22×D411yescharacteristic
372Q8—22no—
373Q8—22no—
374Q8—42no—
375Q8—42no—
376Q8—82no—
377Q8—82no—
378D4—22no—
379D4—22no—
380D4—22no—
381D4—22no—
382D4—22no—
383D4—22no—
384D4—44no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—82no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4C22×D412yes—
412C2×C4C22×D412yes—
413C2×C4C22×D412yes—
414C2×C4C22×D411yescharacteristic
415C2×C4C22×D412yes—
416C2×C4C22×D412yes—
417C2×C4C22×D412yes—
418C2×C4C2511yescharacteristic
419C8—21no—
420C8—22no—
421C8—21no—
422C8—22no—
423C8—21no—
424C82−1+412yes—
425C22—22no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22C23×D411yescharacteristic
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4C2×2−1+411yescharacteristic
450C4#58511yescharacteristic
451C2—41no—
452C2—82no—
453C2—82no—
454C2—82no—
455C2#349711yescharacteristic
456C2#358512yes—
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3557C2semidirect
2. #3548C2semidirect
3. #3399C2semidirect
4. #3397C2semidirect
5. #3389C2semidirect
6. #3388C2semidirect
7. #3376C2semidirect
8. #3370C2semidirect
9. #3368C2semidirect
10. #3366C2semidirect
11. #3260C2semidirect
12. #3259C2semidirect
13. #3257C2semidirect
14. #3251C2semidirect
15. #3180C2semidirect
16. #3179C2semidirect
17. #3176C2semidirect
18. #3174C2semidirect
19. #2961C2semidirect

Maximal subgroups

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

subgroupindex
#35572
#35482
#33992
#33972
#33892
#33882
#33762
#33702
#33682
#33662
#32602
#32592
#32572
#32512
#31802
#31792
#31762
#31742
#29612