Finite Group Databasebeta

← #211957

Subgroups — #211957

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19129C211yesmaximal, characteristic
2#17980C211yesmaximal, characteristic, complemented
3#17916C211yesmaximal, characteristic
4#17913C211yesmaximal, characteristic
5#17316C211yesmaximal, characteristic
6#9725C211yesmaximal, characteristic, complemented
7#9633C211yesmaximal, characteristic, complemented
8#9585C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic
11#9212C211yesmaximal, characteristic
12#8670C211yesmaximal, characteristic
13#8296C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8242C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3107C2211yescharacteristic
18#3107C2211yescharacteristic
19C8×#41C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2960C2211yescharacteristic
23C4×#131C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1680C411yescharacteristic
26#1680C411yescharacteristic
27#1679C411yescharacteristic
28#1679C411yescharacteristic
29#1674C411yescharacteristic
30#1673C411yescharacteristic
31#1670C411yescharacteristic
32#1669C411yescharacteristic
33#1587—21no—
34#1587—21no—
35#1587—21no—
36#1587—21no—
37#1587—21no—
38#1585—21no—
39#1584—21no—
40#1582—21no—
41#1579—21no—
42SD16⋊C8—21no—
43C8×Q16—21no—
44C8×SD16—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1566C2211yescharacteristic
48#1564C2211yescharacteristic
49#1544C2211yescharacteristic
50#1544C2211yescharacteristic
51#1526C2211yescharacteristic
52#1526C2211yescharacteristic
53#1493C2211yescharacteristic
54#1493C2211yescharacteristic
55#1489C2211yescharacteristic
56#1489C2211yescharacteristic
57#1485C2211yescharacteristic
58#1485C2211yescharacteristic
59#1451C2211yescharacteristic
60#1450—21no—
61#1450C2211yescharacteristic
62C8×M16—21no—
63C8×M16—21no—
64C8×M16C2211yescharacteristic
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C2×#322—21no—
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76#533C2311yescharacteristic
77#533C2311yescharacteristic
78C4×#41C2311yescharacteristic
79#500C2×C411yescharacteristic
80#494C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#485—21no—
84#485—21no—
85#475C2×C411yescharacteristic
86#475C2×C411yescharacteristic
87#474C2×C411yescharacteristic
88#474C2×C411yescharacteristic
89#467—21no—
90#467—21no—
91#462C2×C411yescharacteristic
92#462C2×C411yescharacteristic
93#461C2×C411yescharacteristic
94#461C2×C411yescharacteristic
95#452—21no—
96#452—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8C2×C411yescharacteristic
106C8×Q8C2×C411yescharacteristic
107C8×Q8D411yescharacteristic
108C8×Q8D411yescharacteristic
109C4×Q16—21no—
110C4×Q16—21no—
111C4×Q16—21no—
112C4×Q16—21no—
113C4×SD16—21no—
114C4×SD16—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4C2×C411yescharacteristic
120C8×D4C2×C411yescharacteristic
121C8×D4D411yescharacteristic
122C8×D4D411yescharacteristic
123#431—21no—
124#431—21no—
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129C2×#108—21no—
130#424—21no—
131#424—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134#405—21no—
135#405—22no—
136#405—21no—
137#405—21no—
138C4×M16—21no—
139C4×M16C2311yescharacteristic
140C4×M16C2311yescharacteristic
141C4×M16C2311yescharacteristic
142C2×C4×C8—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#341—21no—
149#335—21no—
150#335—21no—
151#335C2×C411yescharacteristic
152#335C2×C411yescharacteristic
153#334—21no—
154#334—21no—
155#334—21no—
156#334—21no—
157#334C2×C411yescharacteristic
158#334C2×C411yescharacteristic
159#332C2×C411yescharacteristic
160#332C2×C411yescharacteristic
161#332C2×C411yescharacteristic
162#332C2×C411yescharacteristic
163#332C2×C411yescharacteristic
164#332C2×C411yescharacteristic
165#332C2×C411yescharacteristic
166#332C2×C411yescharacteristic
167#330—22no—
168#330—22no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322D411yescharacteristic
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191C82—21no—
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41#4111yescharacteristic
196C2×Q16—21no—
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×SD16—21no—
201C2×SD16—21no—
202#131—24no—
203#131—24no—
204C2×M16—22no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8#4111yescharacteristic
217C22×C8#4111yescharacteristic
218C22×C8#4111yescharacteristic
219C22×C8#4111yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224#125—21no—
225#125—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C22×C411yescharacteristic
235C4×Q8C2×D411yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C22×C411yescharacteristic
239C4×D4C2×D411yescharacteristic
240#117—21no—
241#117—21no—
242#117—21no—
243#117#4111yescharacteristic
244C2×C42—21no—
245C2×C42C2411yescharacteristic
246#108—42no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#106—21no—
252#106—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273#103—21no—
274#103—21no—
275#103—21no—
276#103—21no—
277#103—21no—
278#103—21no—
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#102—21no—
284#102—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293#97—22no—
294#97—21no—
295#97—21no—
296#97—41no—
297#97—41no—
298#97#4112yes—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—41no—
313C4×C8—41no—
314C4×C8C22×C411yescharacteristic
315C4×C8C22×C411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324#41—22no—
325#41—22no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8C4×D411yescharacteristic
331C2×D4—21no—
332C2×D4C4×D411yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339Q16—42no—
340Q16—42no—
341Q16—81no—
342Q16—81no—
343SD16—42no—
344SD16—42no—
345M16—28no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—81no—
353M16—81no—
354C2×C8—28no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—81no—
377C2×C8—81no—
378C2×C8—81no—
379C2×C8—81no—
380C2×C8C4×D411yescharacteristic
381C2×C8C4×D411yescharacteristic
382C2×C8C4×D411yescharacteristic
383C2×C8C4×D411yescharacteristic
384C2×C8#12111yescharacteristic
385C2×C8#12111yescharacteristic
386C2×C8#12111yescharacteristic
387C2×C8#12111yescharacteristic
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4C4×D411yescharacteristic
403C4⋊C4C4×D411yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—41no—
413C42—41no—
414C42C23×C411yescharacteristic
415C42C22×D411yescharacteristic
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C23—21no—
419C23#52911yescharacteristic
420Q8—21no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8#44312yes—
430D4—24no—
431D4—21no—
432D4#44312yes—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C4×#4111yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4#52511yescharacteristic
456C2×C4#53211yescharacteristic
457C8—22no—
458C8—24no—
459C8—22no—
460C8—21no—
461C8—21no—
462C8—21no—
463C8—21no—
464C8—21no—
465C8—21no—
466C8—21no—
467C8—21no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—81no—
478C8—81no—
479C8—81no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—41no—
485C22—41no—
486C22C4×#12111yescharacteristic
487C4—21no—
488C4—21no—
489C4—22no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4C2×#44311yescharacteristic
498C4C2×#44311yescharacteristic
499C4#295611yescharacteristic
500C4#295711yescharacteristic
501C4#306911yescharacteristic
502C4#306911yescharacteristic
503C2—41no—
504C2—42no—
505C2#2151911yescharacteristic
506C2#2154411yescharacteristic
507C2#2155611yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41#41
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17980C2semidirect
6. #9725C2semidirect
7. #9633C2semidirect
8. #9585C2semidirect
9. #9585C2semidirect
13. #8296C2semidirect
14. #8246C2semidirect
15. #8242C2semidirect

Maximal subgroups

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

subgroupindex
#191292
#179802
#179162
#179132
#173162
#97252
#96332
#95852
#92122
#86702
#82962
#82462
#82422