Finite Group Databasebeta

← #212007

Subgroups — #212007

Order 512.

528 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#17977C211yesmaximal, characteristic, complemented
3#17916C211yesmaximal, characteristic
4#17913C211yesmaximal, characteristic
5#17330C211yesmaximal, characteristic
6#9725C211yesmaximal, characteristic, complemented
7#9634C211yesmaximal, characteristic, complemented
8#9585C211yesmaximal, characteristic, complemented
9#9585C211yesmaximal, characteristic, complemented
10#9214C211yesmaximal, characteristic
11#9214C211yesmaximal, characteristic
12#8673C211yesmaximal, characteristic
13#8297C211yesmaximal, characteristic, complemented
14#8246C211yesmaximal, characteristic, complemented
15#8244C211yesmaximal, 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#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#1454—21no—
60#1453—21no—
61#1451—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450—21no—
65#1450C2211yescharacteristic
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#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#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
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—
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—22no—
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—21no—
364C2×C8—22no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
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—41no—
461C8—41no—
462C8—42no—
463C8—42no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8#47312yes—
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
497C4C2×#47311yescharacteristic
498C4#306711yescharacteristic
499C2—41no—
500C2—81no—
501C2#2151911yescharacteristic
502C2#2155011yescharacteristic
503C2C4×#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. #9725C2semidirect
7. #9634C2semidirect
8. #9585C2semidirect
9. #9585C2semidirect
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
#191612
#179772
#179162
#179132
#173302
#97252
#96342
#95852
#92142
#86732
#82972
#82462
#82442