Finite Group Databasebeta

← #498192

Subgroups — #498192

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33436C211yesmaximal, characteristic
2#33422C211yesmaximal, characteristic, complemented
3#21592C211yesmaximal, characteristic
4#13162C211yesmaximal, characteristic, complemented
5#13125C211yesmaximal, characteristic, complemented
6#13122C211yesmaximal, characteristic, complemented
7#13106C211yesmaximal, characteristic, complemented
8#13106C211yesmaximal, characteristic, complemented
9#12983C211yesmaximal, characteristic
10#12917C211yesmaximal, characteristic, complemented
11#12914C211yesmaximal, characteristic, complemented
12#12665C211yesmaximal, characteristic, complemented
13#12615C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic, complemented
15C2×#1379C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19#3091C411yescharacteristic
20#2929C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2169—21no—
23#2168—21no—
24#2168—21no—
25D4×C16—21no—
26#2164—21no—
27#2164—21no—
28#2163—21no—
29#2162—21no—
30C2×#364C2211yescharacteristic
31C2×#364C2211yescharacteristic
32#2151C2211yescharacteristic
33#2151C2211yescharacteristic
34#2151C2211yescharacteristic
35C2×#350C2211yescharacteristic
36C2×#350C2211yescharacteristic
37#2113C2211yescharacteristic
38#2112C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2108C2211yescharacteristic
41#2108C2211yescharacteristic
42#2108C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×#346C2211yescharacteristic
45#1929C411yescharacteristic
46#1929C411yescharacteristic
47#1922C411yescharacteristic
48#1922C411yescharacteristic
49#1871C411yescharacteristic
50#1855C411yescharacteristic
51#1826C411yescharacteristic
52#1772C2211yescharacteristic
53#1772C2211yescharacteristic
54#1764C2211yescharacteristic
55#1764C2211yescharacteristic
56C8×C22⋊C4C2211yescharacteristic
57#1379C2211yescharacteristic
58#1379C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1379C2211yescharacteristic
62#1379C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#131C2311yescharacteristic
68C22×M16C2×C411yescharacteristic
69C23×C8C2311yescharacteristic
70#525C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72#504—22no—
73#504—41no—
74C2×M32—21no—
75C2×M32—21no—
76C2×M32—22no—
77C2×M32—41no—
78C2×M32C2311yescharacteristic
79C2×M32C2311yescharacteristic
80C2×M32C2311yescharacteristic
81C2×M32C2311yescharacteristic
82C22×C16—21no—
83C22×C16—21no—
84C22×C16—21no—
85C22×C16—21no—
86C22×C16—22no—
87C22×C16—41no—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#485C811yescharacteristic
93#485C811yescharacteristic
94#484C811yescharacteristic
95#483C811yescharacteristic
96#482C811yescharacteristic
97#482C811yescharacteristic
98#481C811yescharacteristic
99#480C811yescharacteristic
100#435—21no—
101#435—21no—
102C8×D4—21no—
103C8×D4—21no—
104C2×#108C2×C411yescharacteristic
105C2×#108C2×C411yescharacteristic
106#427C2×C411yescharacteristic
107#427C2×C411yescharacteristic
108#424C2×C411yescharacteristic
109C2×#104C2×C411yescharacteristic
110C2×#104C2×C411yescharacteristic
111#416C2×C411yescharacteristic
112#416C2×C411yescharacteristic
113#408C2×C411yescharacteristic
114#407C2×C411yescharacteristic
115C2×C22⋊C8C2×C411yescharacteristic
116#405—21no—
117#405—21no—
118#405C2×C411yescharacteristic
119#405C2×C411yescharacteristic
120#405C2311yescharacteristic
121#405C2311yescharacteristic
122#405C2311yescharacteristic
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C4×C22⋊C4C2×C411yescharacteristic
128#364—22no—
129#364—41no—
130C4⋊C16—21no—
131C4⋊C16—21no—
132C4⋊C16—21no—
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16—21no—
136#350—22no—
137#350—41no—
138C22⋊C16—21no—
139C22⋊C16—21no—
140C22⋊C16—21no—
141C22⋊C16—21no—
142C22⋊C16D411yescharacteristic
143C22⋊C16D411yescharacteristic
144C22⋊C16D411yescharacteristic
145C22⋊C16D411yescharacteristic
146#346—21no—
147#346—21no—
148#346—21no—
149#346—22no—
150#346—41no—
151C4×C16—21no—
152C4×C16—21no—
153C4×C16—21no—
154#341C2×C411yescharacteristic
155#341C2×C411yescharacteristic
156#339C2×C411yescharacteristic
157#339C2×C411yescharacteristic
158#339C2×C411yescharacteristic
159#339C2×C411yescharacteristic
160#339C2×C411yescharacteristic
161#339C2×C411yescharacteristic
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164C2×#41C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166#131—21no—
167#131—21no—
168#131—21no—
169#131—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—41no—
174C2×M16C22×C411yescharacteristic
175C2×M16C22×C411yescharacteristic
176C2×M16C22×C411yescharacteristic
177C2×M16C22×C411yescharacteristic
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8#4111yescharacteristic
190C22×C8#4111yescharacteristic
191C22×C8C2411yescharacteristic
192#123—21no—
193C22⋊Q8C2×C811yescharacteristic
194C22⋊Q8C2×C811yescharacteristic
195#121C2×C811yescharacteristic
196#121C2×C811yescharacteristic
197#120—21no—
198C4×D4—21no—
199C4×D4—21no—
200#117—21no—
201#117—21no—
202#117C2×C811yescharacteristic
203#117C2×C811yescharacteristic
204#117C22×C411yescharacteristic
205#117C22×C411yescharacteristic
206#117C22×C411yescharacteristic
207C2×C4⋊C4C2×C811yescharacteristic
208C2×C4⋊C4C2×C811yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214M32—21no—
215M32—21no—
216M32—21no—
217M32—21no—
218M32—21no—
219M32—21no—
220M32—21no—
221M32—42no—
222M32—42no—
223M32#4112yes—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—42no—
238C2×C16—42no—
239C2×C16—42no—
240C2×C16—42no—
241C2×C16C2×D411yescharacteristic
242C2×C16C2×D411yescharacteristic
243C2×C16C2×D411yescharacteristic
244C2×C16C2×D411yescharacteristic
245C2×C16#4112yes—
246#108—22no—
247#108—41no—
248#107C2×C811yescharacteristic
249#107C2×C811yescharacteristic
250#107C2×C811yescharacteristic
251#107C2×C811yescharacteristic
252#106C2×C811yescharacteristic
253#106C2×C811yescharacteristic
254#106C2×C811yescharacteristic
255#106C2×C811yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258#104—22no—
259#104—41no—
260#103C2×C811yescharacteristic
261#103C2×C811yescharacteristic
262#103C2×C811yescharacteristic
263#103C2×C811yescharacteristic
264#102C2×C811yescharacteristic
265#102C2×C811yescharacteristic
266#102C2×C811yescharacteristic
267#102C2×C811yescharacteristic
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8C2×C811yescharacteristic
273C22⋊C8C2×C811yescharacteristic
274C22⋊C8C2×C811yescharacteristic
275C22⋊C8C2×C811yescharacteristic
276#97—21no—
277#97—21no—
278#97—21no—
279#97—21no—
280#97—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290#95—21no—
291C24#13111yescharacteristic
292#41—21no—
293#41—21no—
294#41—21no—
295#41—21no—
296C2×Q8C22×C811yescharacteristic
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C22×C811yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C22×C811yescharacteristic
313C22×C4C22×C811yescharacteristic
314C22×C4#13111yescharacteristic
315C22×C4C23×C411yescharacteristic
316M16—21no—
317M16—21no—
318M16—21no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—21no—
323M16—42no—
324M16—41no—
325M16—41no—
326M16#13112yes—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8C4×D411yescharacteristic
349C2×C8C4×D411yescharacteristic
350C2×C8C22×C811yescharacteristic
351C2×C8C22×C811yescharacteristic
352C2×C8C22×C811yescharacteristic
353C2×C8C22×C811yescharacteristic
354C2×C8C22×D411yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8C2×#4111yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C82−1+411yescharacteristic
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4C22×C811yescharacteristic
365C4⋊C4C22×C811yescharacteristic
366C4⋊C4C22×C811yescharacteristic
367C4⋊C4C22×C811yescharacteristic
368C4⋊C4C22×C811yescharacteristic
369C4⋊C4C22×C811yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—42no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C16—21no—
390C16—21no—
391C16—22no—
392C16—21no—
393C16—21no—
394C16—22no—
395C16—41no—
396C16—41no—
397C16—41no—
398C16—41no—
399C23—22no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23C8×D411yescharacteristic
405C23C8×D411yescharacteristic
406C23C2×#13111yescharacteristic
407Q8—21no—
408Q8—21no—
409D4—21no—
410D4—21no—
411D4—42no—
412D4—41no—
413D4—41no—
414D4—81no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4C8×D411yescharacteristic
443C2×C4C8×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C4×#4111yescharacteristic
446C2×C4#52011yescharacteristic
447C2×C4C23×C811yescharacteristic
448C2×C4C2×#13111yescharacteristic
449C2×C4#56811yescharacteristic
450C8—21no—
451C8—21no—
452C8—22no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8#54611yescharacteristic
461C8#54711yescharacteristic
462C8#55111yescharacteristic
463C8#55211yescharacteristic
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22#217112yes—
477C22C2×C8×D411yescharacteristic
478C22#296811yescharacteristic
479C22#299411yescharacteristic
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4#235911yescharacteristic
492C4#293311yescharacteristic
493C4C8×#4111yescharacteristic
494C4#298811yescharacteristic
495C2—21no—
496C2—21no—
497C2—22no—
498C2—41no—
499C2—81no—
500C2#2290011yescharacteristic
501C2C2×#217111yescharacteristic
502C2#3344211yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#525C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C22⋊C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33422C2semidirect
4. #13162C2semidirect
5. #13125C2semidirect
6. #13122C2semidirect
7. #13106C2semidirect
8. #13106C2semidirect
10. #12917C2semidirect
11. #12914C2semidirect
12. #12665C2semidirect
13. #12615C2semidirect
14. #12611C2semidirect
15. C2×#1379C2semidirect

Maximal subgroups

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

subgroupindex
#334362
#334222
#215922
#131622
#131252
#131222
#131062
#129832
#129172
#129142
#126652
#126152
#126112
C2×#13792