Finite Group Databasebeta

← #211997

Subgroups — #211997

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19161C211yesmaximal, characteristic
2#17980C211yesmaximal, characteristic, complemented
3#17916C211yesmaximal, characteristic
4#17913C211yesmaximal, characteristic
5#17329C211yesmaximal, characteristic
6#9727C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9587C211yesmaximal, characteristic, complemented
9#9587C211yesmaximal, characteristic, complemented
10#9214C211yesmaximal, characteristic
11#9214C211yesmaximal, characteristic
12#8672C211yesmaximal, characteristic
13#8296C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8242C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3107C2211yescharacteristic
18#3107C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2960C2211yescharacteristic
23#2922C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1700C411yescharacteristic
26#1700C411yescharacteristic
27#1699C411yescharacteristic
28#1699C411yescharacteristic
29#1694C411yescharacteristic
30#1693C411yescharacteristic
31#1690C411yescharacteristic
32#1689C411yescharacteristic
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#1454—21no—
60#1453—21no—
61#1452—21no—
62#1452C2211yescharacteristic
63#1451—21no—
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75#533C2311yescharacteristic
76#533C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#500C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#485—21no—
83#485—21no—
84#475C2×C411yescharacteristic
85#475C2×C411yescharacteristic
86#474C2×C411yescharacteristic
87#474C2×C411yescharacteristic
88#467—21no—
89#467—21no—
90#462C2×C411yescharacteristic
91#462C2×C411yescharacteristic
92#461C2×C411yescharacteristic
93#461C2×C411yescharacteristic
94#452—21no—
95#452—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446C2×C411yescharacteristic
101#446C2×C411yescharacteristic
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8D411yescharacteristic
107C8×Q8D411yescharacteristic
108C4×Q16—21no—
109C4×Q16—21no—
110C4×Q16—21no—
111C4×Q16—21no—
112C4×SD16—21no—
113C4×SD16—21no—
114#436C2×C411yescharacteristic
115#436C2×C411yescharacteristic
116#435—21no—
117#435—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4D411yescharacteristic
121C8×D4D411yescharacteristic
122#433—22no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431—21no—
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131C2×#108—21no—
132#424—21no—
133#424—21no—
134#424—21no—
135#423C2311yescharacteristic
136#423C2311yescharacteristic
137#408—21no—
138#405—21no—
139#405—21no—
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
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#322—21no—
188#322—21no—
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
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—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8#4111yescharacteristic
213C22×C8#4111yescharacteristic
214C22×C8#4111yescharacteristic
215C22×C8#4111yescharacteristic
216C4⋊Q8C22×C411yescharacteristic
217C4⋊Q8C22×C411yescharacteristic
218C4⋊Q8C22×C411yescharacteristic
219C4⋊Q8C22×C411yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8C22×C411yescharacteristic
231C4×Q8C2×D411yescharacteristic
232C4×D4—21no—
233C4×D4—21no—
234C4×D4C22×C411yescharacteristic
235C4×D4C2×D411yescharacteristic
236#117—21no—
237#117—21no—
238#117—21no—
239#117#4111yescharacteristic
240C2×C42—21no—
241C2×C42C2411yescharacteristic
242#108—42no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#106—21no—
248#106—21no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8C22×C411yescharacteristic
264C4⋊C8C22×C411yescharacteristic
265C4⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C2×D411yescharacteristic
270C4⋊C8C2×D411yescharacteristic
271#103—21no—
272#103—21no—
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#102—21no—
282#102—21no—
283C22⋊C8—22no—
284C22⋊C8—22no—
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—21no—
294#97—21no—
295#97—21no—
296#97—21no—
297#97—41no—
298#97—41no—
299#97#4112yes—
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—41no—
311C4×C8—41no—
312C4×C8C22×C411yescharacteristic
313C4×C8C22×C411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322#41—41no—
323#41—41no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C4×D411yescharacteristic
329C2×D4—21no—
330C2×D4C4×D411yescharacteristic
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337Q16—42no—
338Q16—42no—
339Q16—81no—
340Q16—81no—
341SD16—42no—
342SD16—42no—
343M16—22no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—81no—
350M16—81no—
351M16—81no—
352M16—81no—
353C2×C8—21no—
354C2×C8—21no—
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—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—81no—
375C2×C8—81no—
376C2×C8C4×D411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C2×C8C4×D411yescharacteristic
379C2×C8C4×D411yescharacteristic
380C2×C8#12111yescharacteristic
381C2×C8#12111yescharacteristic
382C2×C8#12111yescharacteristic
383C2×C8#12111yescharacteristic
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4C4×D411yescharacteristic
399C4⋊C4C4×D411yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—41no—
409C42—41no—
410C42C23×C411yescharacteristic
411C42C22×D411yescharacteristic
412C42C22×D411yescharacteristic
413C42C2×#4111yescharacteristic
414C23—21no—
415C23#52911yescharacteristic
416Q8—21no—
417Q8—21no—
418Q8—41no—
419Q8—41no—
420Q8—41no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426D4—21no—
427D4—21no—
428D4—42no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C4×#4111yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4#53211yescharacteristic
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—21no—
458C8—21no—
459C8—21no—
460C8—21no—
461C8—41no—
462C8—41no—
463C8—42no—
464C8—42no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22C4×#12111yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4#295511yescharacteristic
494C4#295511yescharacteristic
495C4#295611yescharacteristic
496C4#295711yescharacteristic
497C4#306911yescharacteristic
498C4#306911yescharacteristic
499C2—41no—
500C2—81no—
501C2#2151911yescharacteristic
502C2#2154811yescharacteristic
503C2#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. #9727C2semidirect
7. #9635C2semidirect
8. #9587C2semidirect
9. #9587C2semidirect
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
#191612
#179802
#179162
#179132
#173292
#97272
#96352
#95872
#92142
#86722
#82962
#82462
#82422