Finite Group Databasebeta

← #211967

Subgroups — #211967

Order 512.

551 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#17977C211yesmaximal, characteristic, complemented
3#17916C211yesmaximal, characteristic
4#17913C211yesmaximal, characteristic
5#17320C211yesmaximal, characteristic
6#9727C211yesmaximal, characteristic, complemented
7#9636C211yesmaximal, characteristic, complemented
8#9587C211yesmaximal, characteristic, complemented
9#9587C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic
11#9212C211yesmaximal, characteristic
12#8671C211yesmaximal, characteristic
13#8297C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8244C211yesmaximal, 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#1585—21no—
35#1584—21no—
36#1584—21no—
37#1584—21no—
38#1584—21no—
39#1584—21no—
40#1582—21no—
41#1579—21no—
42SD16⋊C8—21no—
43C8×Q16—21no—
44C8×SD16—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1567C2211yescharacteristic
48#1564C2211yescharacteristic
49#1544C2211yescharacteristic
50#1544C2211yescharacteristic
51#1526C2211yescharacteristic
52#1526C2211yescharacteristic
53#1493C2211yescharacteristic
54#1493C2211yescharacteristic
55#1489C2211yescharacteristic
56#1489C2211yescharacteristic
57#1485C2211yescharacteristic
58#1485C2211yescharacteristic
59#1452—21no—
60#1452C2211yescharacteristic
61#1451—21no—
62#1451—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
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#335—21no—
152#335—21no—
153#335C2×C411yescharacteristic
154#335C2×C411yescharacteristic
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#322—21no—
188#322—21no—
189#322—21no—
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193#322D411yescharacteristic
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—42no—
203#131—42no—
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—22no—
346M16—41no—
347M16—44no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—81no—
353M16—81no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—22no—
365C2×C8—21no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
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#44412yes—
430D4—24no—
431D4—21no—
432D4#44412yes—
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—21no—
458C8—21no—
459C8—21no—
460C8—21no—
461C8—21no—
462C8—21no—
463C8—21no—
464C8—41no—
465C8—42no—
466C8—41no—
467C8—42no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8#47312yes—
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×#44411yescharacteristic
498C4C2×#44411yescharacteristic
499C4#295611yescharacteristic
500C4#295711yescharacteristic
501C4C2×#47311yescharacteristic
502C4#306711yescharacteristic
503C2—41no—
504C2—42no—
505C2#2151911yescharacteristic
506C2#2154611yescharacteristic
507C2C4×#47311yescharacteristic
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. #17977C2semidirect
6. #9727C2semidirect
7. #9636C2semidirect
8. #9587C2semidirect
9. #9587C2semidirect
13. #8297C2semidirect
14. #8246C2semidirect
15. #8244C2semidirect

Maximal subgroups

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

subgroupindex
#191292
#179772
#179162
#179132
#173202
#97272
#96362
#95872
#92122
#86712
#82972
#82462
#82442