Finite Group Databasebeta

← #498596

Subgroups — #498596

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33476C211yesmaximal, characteristic, complemented
2#33438C211yesmaximal, characteristic
3#21661C211yesmaximal, characteristic
4#13265C211yesmaximal, characteristic, complemented
5#13193C211yesmaximal, characteristic
6#13125C211yesmaximal, characteristic, complemented
7#13124C211yesmaximal, characteristic, complemented
8#13122C211yesmaximal, characteristic, complemented
9#13122C211yesmaximal, characteristic, complemented
10#12917C211yesmaximal, characteristic, complemented
11#12917C211yesmaximal, characteristic, complemented
12#12656C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12615C211yesmaximal, characteristic, complemented
15#12614C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18#3135C411yescharacteristic
19#2960C2211yescharacteristic
20#2931C2211yescharacteristic
21#2183—21no—
22#2183—21no—
23#2183—21no—
24Q8×C16—21no—
25#2169—21no—
26#2168—21no—
27#2168—21no—
28D4×C16—21no—
29C2×#364C2211yescharacteristic
30C2×#364C2211yescharacteristic
31#2151C2211yescharacteristic
32#2151C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×#350C2211yescharacteristic
36C2×#350C2211yescharacteristic
37C2×#350C2211yescharacteristic
38C2×#350C2211yescharacteristic
39#2113C2211yescharacteristic
40#2113C2211yescharacteristic
41#2108C2211yescharacteristic
42#2108C2211yescharacteristic
43C4×M32C2211yescharacteristic
44C4×M32C2211yescharacteristic
45C2×#346C2211yescharacteristic
46C2×C4×C16C2211yescharacteristic
47#1995C411yescharacteristic
48#1963C411yescharacteristic
49#1938C411yescharacteristic
50#1938C411yescharacteristic
51#1938C411yescharacteristic
52#1938C411yescharacteristic
53#1838C411yescharacteristic
54#1764C2211yescharacteristic
55#1764C2211yescharacteristic
56#1764C2211yescharacteristic
57#1764C2211yescharacteristic
58#1726C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1379C2211yescharacteristic
62#1379C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#131C2311yescharacteristic
68C2×#131C2311yescharacteristic
69#533C2×C411yescharacteristic
70#532C2×C411yescharacteristic
71#504—22no—
72#504—22no—
73#504—41no—
74#504—41no—
75C2×M32—21no—
76C2×M32—21no—
77C2×M32C2311yescharacteristic
78C2×M32C2311yescharacteristic
79C2×M32C2311yescharacteristic
80C2×M32C2311yescharacteristic
81C22×C16—21no—
82C22×C16—21no—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#491C811yescharacteristic
88#491C811yescharacteristic
89#490C811yescharacteristic
90#490C811yescharacteristic
91#489C811yescharacteristic
92#489C811yescharacteristic
93#489C811yescharacteristic
94#489C811yescharacteristic
95#446—21no—
96#446—21no—
97C8×Q8—21no—
98C8×Q8—21no—
99#436—21no—
100#436—21no—
101C8×D4—21no—
102C8×D4—21no—
103#424C2×C411yescharacteristic
104#424C2×C411yescharacteristic
105#423C2×C411yescharacteristic
106#423C2×C411yescharacteristic
107C2×#104C2×C411yescharacteristic
108C2×#104C2×C411yescharacteristic
109C2×#104C2×C411yescharacteristic
110C2×#104C2×C411yescharacteristic
111#416C2×C411yescharacteristic
112#416C2×C411yescharacteristic
113#416C2×C411yescharacteristic
114#416C2×C411yescharacteristic
115#408C2×C411yescharacteristic
116#408C2×C411yescharacteristic
117#405C2311yescharacteristic
118#405C2311yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2×C411yescharacteristic
121C2×#97C2×C411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#376C2×C411yescharacteristic
126#364—22no—
127#364—22no—
128C4⋊C16—21no—
129C4⋊C16—21no—
130C4⋊C16—21no—
131C4⋊C16—21no—
132C4⋊C16—21no—
133C4⋊C16—21no—
134C4⋊C16D411yescharacteristic
135C4⋊C16D411yescharacteristic
136C4⋊C16D411yescharacteristic
137C4⋊C16D411yescharacteristic
138#350—22no—
139#350—22no—
140#350—41no—
141#350—41no—
142C22⋊C16—21no—
143C22⋊C16—21no—
144C22⋊C16—21no—
145C22⋊C16—21no—
146#346—21no—
147#346—21no—
148#346—21no—
149#346—21no—
150#346—41no—
151C4×C16—21no—
152C4×C16—21no—
153C4×C16—21no—
154C4×C16—21no—
155C4×C16—41no—
156#341C2×C411yescharacteristic
157#341C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#341C2×C411yescharacteristic
160#339C2×C411yescharacteristic
161#339C2×C411yescharacteristic
162#339C2×C411yescharacteristic
163#339C2×C411yescharacteristic
164C2×#41C22×C411yescharacteristic
165C2×#41C22×C411yescharacteristic
166#131—21no—
167#131—21no—
168#131—21no—
169#131—21no—
170#131—21no—
171#131—21no—
172#131—21no—
173#131—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16C22×C411yescharacteristic
177C2×M16C22×C411yescharacteristic
178C2×M16C22×C411yescharacteristic
179C2×M16C22×C411yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C2411yescharacteristic
186C4⋊Q8C2×C811yescharacteristic
187C4⋊Q8C2×C811yescharacteristic
188C4⋊Q8C2×C811yescharacteristic
189#127C2×C811yescharacteristic
190#125C2×C811yescharacteristic
191#125C2×C811yescharacteristic
192#124C2×C811yescharacteristic
193#124C2×C811yescharacteristic
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#121—21no—
197#121—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117C22×C411yescharacteristic
203#117C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32—42no—
214M32—42no—
215M32#4112yes—
216M32#4112yes—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—42no—
232C2×C16—42no—
233C2×C16C2×D411yescharacteristic
234C2×C16C2×D411yescharacteristic
235C2×C16C2×D411yescharacteristic
236C2×C16C2×D411yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245#104—22no—
246#104—22no—
247#104—41no—
248#104—41no—
249#103C2×C811yescharacteristic
250#103C2×C811yescharacteristic
251#103C2×C811yescharacteristic
252#103C2×C811yescharacteristic
253#103C2×C811yescharacteristic
254#103C2×C811yescharacteristic
255#103C2×C811yescharacteristic
256#103C2×C811yescharacteristic
257#102C2×C811yescharacteristic
258#102C2×C811yescharacteristic
259#102C2×C811yescharacteristic
260#102C2×C811yescharacteristic
261#102C2×C811yescharacteristic
262#102C2×C811yescharacteristic
263#102C2×C811yescharacteristic
264#102C2×C811yescharacteristic
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269#97—21no—
270#97—21no—
271#97—22no—
272#97—41no—
273#97C2×C811yescharacteristic
274#97C2×C811yescharacteristic
275#97C2×C811yescharacteristic
276#97C2×C811yescharacteristic
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8C2×D411yescharacteristic
284C4×C8C2×D411yescharacteristic
285C4×C8#4111yescharacteristic
286C4×C8#4111yescharacteristic
287#95—21no—
288#95—21no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—21no—
296#41—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8C22×C811yescharacteristic
300C2×Q8C22×C811yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C22×C811yescharacteristic
306C2×D4C22×C811yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4#13111yescharacteristic
312C22×C4#13111yescharacteristic
313C22×C4C23×C411yescharacteristic
314M16—21no—
315M16—21no—
316M16—21no—
317M16—21no—
318M16—21no—
319M16—21no—
320M16—21no—
321M16—21no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326C2×C8—21no—
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—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8C4×D411yescharacteristic
341C2×C8C4×D411yescharacteristic
342C2×C8C22×C811yescharacteristic
343C2×C8C22×C811yescharacteristic
344C2×C8C22×C811yescharacteristic
345C2×C8C22×C811yescharacteristic
346C2×C8C22×D411yescharacteristic
347C2×C8C2×#4111yescharacteristic
348C2×C8C2×#4111yescharacteristic
349C2×C8C2×#4111yescharacteristic
350C2×C8H(5,𝔽2)11yescharacteristic
351C2×C82−1+411yescharacteristic
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4C22×C811yescharacteristic
361C4⋊C4C22×C811yescharacteristic
362C4⋊C4C22×C811yescharacteristic
363C4⋊C4C22×C811yescharacteristic
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42C22×C811yescharacteristic
378C42C22×C811yescharacteristic
379C42#13112yes—
380C16—21no—
381C16—21no—
382C16—21no—
383C16—21no—
384C16—22no—
385C16—22no—
386C16—41no—
387C16—41no—
388C16—41no—
389C16—41no—
390C23—21no—
391C23—21no—
392C23C2×#13111yescharacteristic
393Q8—21no—
394Q8—21no—
395Q8—21no—
396Q8—21no—
397Q8—42no—
398Q8—81no—
399D4—21no—
400D4—21no—
401D4—21no—
402D4—21no—
403D4—42no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—81no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4C8×D411yescharacteristic
428C2×C4C8×D411yescharacteristic
429C2×C4C8×D411yescharacteristic
430C2×C4C8×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C4×#4111yescharacteristic
433C2×C4#52011yescharacteristic
434C2×C4C23×C811yescharacteristic
435C2×C4C2×#13111yescharacteristic
436C2×C4#56811yescharacteristic
437C8—21no—
438C8—21no—
439C8—21no—
440C8—21no—
441C8—21no—
442C8—21no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—41no—
447C8#54611yescharacteristic
448C8#54811yescharacteristic
449C8#55611yescharacteristic
450C8#55911yescharacteristic
451C22—21no—
452C22—21no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22C2×C8×D411yescharacteristic
460C22#296911yescharacteristic
461C22#299411yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—81no—
474C4#236711yescharacteristic
475C4#293511yescharacteristic
476C4C8×#4111yescharacteristic
477C4#298811yescharacteristic
478C2—21no—
479C2—21no—
480C2—81no—
481C2—81no—
482C2#2320411yescharacteristic
483C2#3344111yescharacteristic
484C2#3344211yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#532C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33476C2semidirect
4. #13265C2semidirect
6. #13125C2semidirect
7. #13124C2semidirect
8. #13122C2semidirect
9. #13122C2semidirect
10. #12917C2semidirect
11. #12917C2semidirect
12. #12656C2semidirect
13. #12654C2semidirect
14. #12615C2semidirect
15. #12614C2semidirect

Maximal subgroups

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

subgroupindex
#334762
#334382
#216612
#132652
#131932
#131252
#131242
#131222
#129172
#126562
#126542
#126152
#126142