Finite Group Databasebeta

← #496856

Subgroups — #496856

Order 512.

515 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33433C211yesmaximal, characteristic
2#33397C211yesmaximal, characteristic, complemented
3#21592C211yesmaximal, characteristic
4#13162C211yesmaximal, characteristic, complemented
5#13161C211yesmaximal, characteristic, complemented
6#13125C211yesmaximal, characteristic, complemented
7#13124C211yesmaximal, characteristic, complemented
8#13106C211yesmaximal, characteristic, complemented
9#13106C211yesmaximal, characteristic, complemented
10#12978C211yesmaximal, characteristic
11#12917C211yesmaximal, characteristic, complemented
12#12917C211yesmaximal, characteristic, complemented
13#12914C211yesmaximal, characteristic, complemented
14#12848C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19#3091C411yescharacteristic
20#2929C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2169—21no—
23#2169—21no—
24#2168—21no—
25#2168—21no—
26C2×#364C2211yescharacteristic
27C2×#364C2211yescharacteristic
28#2151—21no—
29#2151—21no—
30#2151C2211yescharacteristic
31#2151C2211yescharacteristic
32#2151C2211yescharacteristic
33#2151C2211yescharacteristic
34#2151C2211yescharacteristic
35#2150—21no—
36#2150—21no—
37C2×#350C2211yescharacteristic
38C2×#350C2211yescharacteristic
39#2113C2211yescharacteristic
40#2112C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42#2108C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
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—41no—
73#504—41no—
74C2×M32—21no—
75C2×M32—21no—
76C2×M32—41no—
77C2×M32—41no—
78C2×M32—41no—
79C2×M32—41no—
80C2×M32D411yescharacteristic
81C2×M32D411yescharacteristic
82C2×M32D411yescharacteristic
83C2×M32D411yescharacteristic
84C2×M32C2311yescharacteristic
85C2×M32C2311yescharacteristic
86C2×M32C2311yescharacteristic
87C2×M32C2311yescharacteristic
88C22×C16—21no—
89C22×C16—21no—
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92C22×C16C2311yescharacteristic
93C22×C16C2311yescharacteristic
94#485C811yescharacteristic
95#485C811yescharacteristic
96#484C811yescharacteristic
97#483C811yescharacteristic
98#482C811yescharacteristic
99#482C811yescharacteristic
100#481C811yescharacteristic
101#480C811yescharacteristic
102#435—21no—
103#435—21no—
104C8×D4—21no—
105C8×D4—21no—
106C2×#108C2×C411yescharacteristic
107C2×#108C2×C411yescharacteristic
108#427C2×C411yescharacteristic
109#427C2×C411yescharacteristic
110#424C2×C411yescharacteristic
111C2×#104C2×C411yescharacteristic
112C2×#104C2×C411yescharacteristic
113#416C2×C411yescharacteristic
114#416C2×C411yescharacteristic
115#408C2×C411yescharacteristic
116#407C2×C411yescharacteristic
117C2×C22⋊C8C2×C411yescharacteristic
118#405—21no—
119#405—21no—
120#405C2×C411yescharacteristic
121#405C2×C411yescharacteristic
122#405C2311yescharacteristic
123#405C2311yescharacteristic
124#405C2311yescharacteristic
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C22⋊C4C2×C411yescharacteristic
130#364—41no—
131#364—41no—
132C4⋊C16—21no—
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16—21no—
136C4⋊C16—21no—
137C4⋊C16—21no—
138C4⋊C16—21no—
139C4⋊C16—21no—
140C4⋊C16—21no—
141C4⋊C16—21no—
142#350—41no—
143#350—41no—
144C22⋊C16—21no—
145C22⋊C16—21no—
146C22⋊C16—21no—
147C22⋊C16—21no—
148C22⋊C16—21no—
149C22⋊C16—21no—
150#346—21no—
151#346—41no—
152C4×C16—21no—
153C4×C16—41no—
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—41no—
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—21no—
222M32—41no—
223M32—41no—
224M32—41no—
225M32—41no—
226M32—42no—
227M32—42no—
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—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—42no—
243C2×C16—42no—
244C2×C16C2×D411yescharacteristic
245C2×C16C2×D411yescharacteristic
246C2×C16C2×D411yescharacteristic
247C2×C16C2×D411yescharacteristic
248#108—41no—
249#108—41no—
250#107C2×C811yescharacteristic
251#107C2×C811yescharacteristic
252#107C2×C811yescharacteristic
253#107C2×C811yescharacteristic
254#106C2×C811yescharacteristic
255#106C2×C811yescharacteristic
256#106C2×C811yescharacteristic
257#106C2×C811yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260#104—41no—
261#104—41no—
262#103C2×C811yescharacteristic
263#103C2×C811yescharacteristic
264#103C2×C811yescharacteristic
265#103C2×C811yescharacteristic
266#102C2×C811yescharacteristic
267#102C2×C811yescharacteristic
268#102C2×C811yescharacteristic
269#102C2×C811yescharacteristic
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8C2×C811yescharacteristic
275C22⋊C8C2×C811yescharacteristic
276C22⋊C8C2×C811yescharacteristic
277C22⋊C8C2×C811yescharacteristic
278#97—21no—
279#97—21no—
280#97—21no—
281#97—21no—
282#97—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292#95—21no—
293C24#13111yescharacteristic
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298C2×Q8C22×C811yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C22×C811yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C22×C811yescharacteristic
315C22×C4C22×C811yescharacteristic
316C22×C4#13111yescharacteristic
317C22×C4C23×C411yescharacteristic
318M16—21no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—21no—
323M16—21no—
324M16—21no—
325M16—21no—
326M16—42no—
327M16—41no—
328M16—41no—
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—21no—
346C2×C8—21no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8C22×C811yescharacteristic
353C2×C8C22×C811yescharacteristic
354C2×C8C22×C811yescharacteristic
355C2×C8C22×C811yescharacteristic
356C2×C8C22×D411yescharacteristic
357C2×C8C2×#4111yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C2×C82−1+411yescharacteristic
361C2×C82−1+411yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4C22×C811yescharacteristic
367C4⋊C4C22×C811yescharacteristic
368C4⋊C4C22×C811yescharacteristic
369C4⋊C4C22×C811yescharacteristic
370C4⋊C4C22×C811yescharacteristic
371C4⋊C4C22×C811yescharacteristic
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—42no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C16—21no—
392C16—21no—
393C16—21no—
394C16—21no—
395C16—41no—
396C16—41no—
397C16—41no—
398C16—41no—
399C16—41no—
400C16—41no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23C8×D411yescharacteristic
407C23C8×D411yescharacteristic
408C23C2×#13111yescharacteristic
409Q8—21no—
410Q8—21no—
411D4—21no—
412D4—21no—
413D4—41no—
414D4—41no—
415D4—81no—
416D4—81no—
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—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4C8×D411yescharacteristic
445C2×C4C8×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4#52011yescharacteristic
448C2×C4#52011yescharacteristic
449C2×C4C23×C811yescharacteristic
450C2×C4#56811yescharacteristic
451C2×C4#56811yescharacteristic
452C8—21no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—21no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8#53411yescharacteristic
463C8#53611yescharacteristic
464C8#53911yescharacteristic
465C8#53911yescharacteristic
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—81no—
478C22—81no—
479C22C2×C8×D411yescharacteristic
480C22#299211yescharacteristic
481C22#299211yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—81no—
493C4#232211yescharacteristic
494C4#293011yescharacteristic
495C4#298811yescharacteristic
496C4#298811yescharacteristic
497C2—21no—
498C2—21no—
499C2—41no—
500C2—41no—
501C2—81no—
502C2#2206311yescharacteristic
503C2#3344211yescharacteristic
504C2#3344311yescharacteristic
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 ◁ C22×M32 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33397C2semidirect
4. #13162C2semidirect
5. #13161C2semidirect
6. #13125C2semidirect
7. #13124C2semidirect
8. #13106C2semidirect
9. #13106C2semidirect
11. #12917C2semidirect
12. #12917C2semidirect
13. #12914C2semidirect
14. #12848C2semidirect
15. #12612C2semidirect

Maximal subgroups

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

subgroupindex
#334332
#333972
#215922
#131622
#131612
#131252
#131242
#131062
#129782
#129172
#129142
#128482
#126122