Finite Group Databasebeta

← #498212

Subgroups — #498212

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C211yesmaximal, characteristic
2#23024C211yesmaximal, characteristic
3#22611C211yesmaximal, characteristic
4#13167C211yesmaximal, characteristic, complemented
5#13150C211yesmaximal, characteristic, complemented
6#13133C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic
10#12936C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic
12#12674C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic
15C2×#1380C211yesmaximal, characteristic
16C23×C16C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2360—21no—
22#2344—21no—
23#2235C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1928—21no—
40#1918—21no—
41#1918—21no—
42#1872—21no—
43#1863—21no—
44#1823—21no—
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C22⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—41no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C16—22no—
68C22×C16—22no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#467—22no—
75#467C2×C414yes—
76C4×Q16—22no—
77C4×Q16—41no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—21no—
82C2×#107—41no—
83C2×#107C2×C414yes—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—41no—
102#387—41no—
103#387—41no—
104#387—41no—
105#385—41no—
106#381—41no—
107#380—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4—41no—
112C4×C22⋊C4D411yescharacteristic
113#367C2×C414yescomplemented
114#367C2×C414yescomplemented
115#366—22no—
116#366—41no—
117#358—22no—
118#358—41no—
119#358C2×C414yescomplemented
120#358C2×C414yescomplemented
121C22⋊C16C2×C414yes—
122#346—22no—
123#346—41no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×Q8—21no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C22×C412yes—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C22×C412yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8#4112yes—
152C22×C8C2411yescharacteristic
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C4×Q8—42no—
156C4×Q8—81no—
157#117—84no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—81no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4D811yescharacteristic
179C2×C22⋊C4D811yescharacteristic
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—44no—
196C2×C16C22×C412yes—
197C2×C16C22×C412yes—
198C2×C16#4112yes—
199#107—24no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—42no—
211#107C22×C412yes—
212#107C22×C412yes—
213#107C22×C412yes—
214#107#4112yes—
215#103—24no—
216#103—42no—
217#103—81no—
218C22⋊C8—22no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95D811yescharacteristic
233#95D811yescharacteristic
234C24C2×D811yescharacteristic
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—161no—
259C22×C4C4×D412yes—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4#13512yes—
266C22×C4C22×D411yescharacteristic
267Q16—22no—
268Q16—22no—
269Q16—41no—
270Q16—41no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8C4×D412yes—
284C2×C8C23×C411yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C2×C82−1+411yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—161no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C42—82no—
321C42—82no—
322C42—161no—
323C16—22no—
324C16—22no—
325C16—22no—
326C16—41no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C4×D812yes—
332C23#50912yes—
333C23#51012yes—
334C23C22×D811yescharacteristic
335Q8—42no—
336Q8—42no—
337Q8—81no—
338Q8—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—88no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—164no—
383C2×C4C4×D812yes—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4#57711yescharacteristic
389C2×C4#57811yescharacteristic
390C8—22no—
391C8—41no—
392C8—41no—
393C8C4×#4111yescharacteristic
394C8#52011yescharacteristic
395C8#55111yescharacteristic
396C8#55211yescharacteristic
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22#217514yes—
404C22C2×C4×D811yescharacteristic
405C22#329411yescharacteristic
406C22#333411yescharacteristic
407C22C2×#50911yescharacteristic
408C22C2×#51011yescharacteristic
409C22#341611yescharacteristic
410C22#341711yescharacteristic
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#235911yescharacteristic
425C4#294711yescharacteristic
426C4#330011yescharacteristic
427C4#333711yescharacteristic
428C2—24no—
429C2#2291611yescharacteristic
430C2C2×#217511yescharacteristic
431C2#3345511yescharacteristic
432C2#3387311yescharacteristic
433C2#3387911yescharacteristic
434C2#3391011yescharacteristic
435C2#3391111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13167C2semidirect
5. #13150C2semidirect
6. #13133C2semidirect
7. #13111C2semidirect
8. #13110C2semidirect
10. #12936C2semidirect
12. #12674C2semidirect
13. #12638C2semidirect
23. #2235C4semidirect
113. #367C2×C4semidirect
114. #367C2×C4semidirect
119. #358C2×C4semidirect
120. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
#230242
#226112
#131672
#131502
#131332
#131112
#131102
#130602
#129362
#129192
#126742
#126382
#126112
C2×#13802