Finite Group Databasebeta

← #6373539

Subgroups — #6373539

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#34209C211yesmaximal, characteristic
4#21647C214yesmaximal, complemented
5#21645C218yesmaximal, complemented
6C4×#487C214yesmaximal, complemented
7#21324C211yesmaximal, characteristic
8#21202C211yesmaximal, characteristic
9#21088C214yesmaximal
10#20760C211yesmaximal, characteristic
11#20417C214yesmaximal
12C4×#405C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#519C2212yes—
16C2×#487C4116yescomplemented
17C2×#487C2212yes—
18C2×#487C2214yes—
19C2×#487C2212yes—
20C2×#419C2214yes—
21C22×#103C2214yes—
22C2×#405C2212yes—
23C2×#405C2212yes—
24C2×C4×M16C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2369—22no—
27#2365—22no—
28#2365—22no—
29#2365—22no—
30#2345—22no—
31#2344—22no—
32#2344—22no—
33Q8×C22⋊C4—22no—
34#2331C2214yes—
35#2327C2214yes—
36#2316C2212yes—
37#2310C2212yes—
38C4×C4⋊Q8C2214yes—
39C4×C4⋊Q8C2214yes—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#1993C2214yes—
45#1990C2214yes—
46#1960C2214yes—
47#1956C2214yes—
48#1935C22116yes—
49#1933C22116yes—
50#1835C2214yes—
51#1834C2212yes—
52#1834C2212yes—
53#1763C22116yes—
54C4×#103C22116yes—
55#1736C2214yes—
56#1735C2214yes—
57#1726C2214yes—
58C4×#97C2212yes—
59C42×C8C2212yes—
60C23×Q8—21no—
61C23×Q8C2312yes—
62C22×M16C2312yes—
63C23×C8C2312yes—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8C2×C418yes—
71C2×C4⋊Q8C2×C418yes—
72C2×C4⋊Q8C2312yes—
73C2×C4⋊Q8C2312yes—
74C2×C4⋊Q8C2312yes—
75C2×C4⋊Q8C2312yes—
76#519—216no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8C2318yes—
80C2×#117—24no—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C2×#117C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2312yes—
89C22×C42C2311yescharacteristic
90#487C2×C4116yes—
91#487C2×C4132yes—
92#487C2×C4116yes—
93#419—216no—
94C2×#103C2×C4132yes—
95C2×#103C23116yes—
96C2×#103C2318yes—
97C2×#103C2318yes—
98#405—28no—
99#405—28no—
100C4×M16—24no—
101C2×#97C2312yes—
102C2×#97C2314yes—
103C2×#97C2314yes—
104C2×C4×C8C2×C418yes—
105C2×C4×C8C2312yes—
106C2×C4×C8C2314yes—
107C2×C4×C8C2314yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2312yes—
111#391—24no—
112#391—24no—
113#391—28no—
114#391C2318yes—
115#389—24no—
116#389—24no—
117#389—24no—
118#389—24no—
119#389C2318yes—
120#387—28no—
121#387—28no—
122#387—28no—
123#384C2312yes—
124#384C2312yes—
125#382C2312yes—
126#382C2312yes—
127#381—24no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D412yes—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2318yes—
144C4×C22⋊C4—24no—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4D412yes—
147#376C2312yes—
148C43C2312yes—
149#340C2318yes—
150#340C2318yes—
151#340C23116yes—
152C22×Q8—28no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—28no—
159C22×Q8—84no—
160C22×Q8C22×C418yes—
161C22×Q8C2414yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—24no—
168C2×M16—24no—
169C2×M16#4118yes—
170C22×C8—24no—
171C22×C8—24no—
172C22×C8C22×C418yes—
173C22×C8C2412yes—
174C22×C8C2412yes—
175C22×C8C2414yes—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8—28no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8C22×C418yes—
185C4⋊Q8C22×C418yes—
186C4⋊Q8C22×C418yes—
187C4⋊Q8C22×C418yes—
188C4×Q8—232no—
189C4×Q8—416no—
190C4×Q8—88no—
191#117—24no—
192#117—28no—
193#117—28no—
194#117—416no—
195#117#4118yes—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4C22×C418yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C42—28no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—28no—
227C2×C42C22×C414yes—
228C2×C42C2×D414yes—
229C2×C42C2×D414yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4114yes—
233C2×C42C2412yes—
234C2×C42C2411yescharacteristic
235C2×C42C2411yescharacteristic
236C2×C42C2412yes—
237#103—232no—
238#103C22×C4132yes—
239#103C22×C4132yes—
240#97—28no—
241#97—24no—
242#97—28no—
243C4×C8—28no—
244C4×C8—24no—
245C4×C8—28no—
246C4×C8C22×C414yes—
247C4×C8C22×C414yes—
248C4×C8C22×C418yes—
249#95—28no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95C2×D414yes—
258#95C2×D414yes—
259C24C22×D411yescharacteristic
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—28no—
263C2×Q8—24no—
264C2×Q8—232no—
265C2×Q8—28no—
266C2×Q8—28no—
267C2×Q8—44no—
268C2×Q8—416no—
269C2×Q8—44no—
270C2×Q8—48no—
271C2×Q8—84no—
272C2×Q8—816no—
273C2×Q8C23×C418yes—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—84no—
285C22×C4C4×D414yes—
286C22×C4C4×D414yes—
287C22×C4C23×C412yes—
288C22×C4C22×D414yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D414yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4114yes—
297C22×C4C2×#4114yes—
298C22×C4C2511yescharacteristic
299M16—216no—
300C2×C8—216no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8C23×C414yes—
308C2×C8C23×C414yes—
309C2×C8C2×#4118yes—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—416no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—88no—
327C4⋊C4#12418yes—
328C4⋊C4C23×C418yes—
329C22⋊C4—24no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4#12418yes—
333C42—28no—
334C42—28no—
335C42—24no—
336C42—44no—
337C42—416no—
338C42—44no—
339C42—88no—
340C42C4×D414yes—
341C42C4×D414yes—
342C42C23×C412yes—
343C42C23×C412yes—
344C42C2×#4112yes—
345C42C2×#4112yes—
346C42C2×#4118yes—
347C23—22no—
348C23—22no—
349C23C2×C4×D412yes—
350C23C2×#12412yes—
351C23#53212yes—
352C23C23×D411yescharacteristic
353Q8—216no—
354Q8—48no—
355Q8—416no—
356Q8—416no—
357Q8—88no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—416no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—88no—
380C2×C4C2×C4×D412yes—
381C2×C4C2×C4×D412yes—
382C2×C4C2×C4×D412yes—
383C2×C4C2×C4×D412yes—
384C2×C4C2×C4×D412yes—
385C2×C4C4×#4118yes—
386C2×C4C2×#12418yes—
387C2×C4C2×#12412yes—
388C2×C4#52918yes—
389C2×C4#53212yes—
390C2×C4#57714yes—
391C2×C4#57814yes—
392C2×C4C24×C411yescharacteristic
393C2×C4C22×#4112yes—
394C2×C4C22×#4112yes—
395C2×C4C22×#4111yescharacteristic
396C8—24no—
397C8—24no—
398C8—28no—
399C22—22no—
400C22—22no—
401C22—24no—
402C22—24no—
403C22C4×#12414yes—
404C22C22×C4×D411yescharacteristic
405C22C22×#12411yescharacteristic
406C22C2×#53211yescharacteristic
407C22C2×#57712yes—
408C22C2×#57812yes—
409C4—22no—
410C4—22no—
411C4—28no—
412C4—48no—
413C4—44no—
414C4—48no—
415C4—84no—
416C4—84no—
417C4C4×#12414yes—
418C4#294914yes—
419C4#295014yes—
420C4C2×C4×#4112yes—
421C4C2×#52912yes—
422C2—24no—
423C2C2×C4×#12411yescharacteristic
424C2C2×#294911yescharacteristic
425C2C2×#295011yescharacteristic
426C2#6062312yes—
427C2#6072312yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21647C2semidirect
5. #21645C2semidirect
6. C4×#487C2semidirect
16. C2×#487C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#342162
#342092
#216472
#216452
C4×#4872
#213242
#212022
#210882
#207602
#204172
C4×#4052