Finite Group Databasebeta

← #108562

Subgroups — #108562

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20525C211yesmaximal, characteristic
2#8009C211yesmaximal, characteristic
3#7943C211yesmaximal, characteristic
4#7942C211yesmaximal, characteristic
5C2×#1284C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7913C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1753—22no—
16#1752—21no—
17C8×C22⋊C4—21no—
18#1749C412yescomplemented
19C4×C22⋊C8C412yescomplemented
20#1293—22no—
21#1284—22no—
22#1284—21no—
23#1284—21no—
24#1284—22no—
25#1282C414yescomplemented
26#1280C414yescomplemented
27C24×C4C2311yescharacteristic
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C23×C8—21no—
32C2×#117—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C42C2×C412yes—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407D412yes—
47#407Q812yes—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8C2×C411yescharacteristic
59C2×C22⋊C8C2×C411yescharacteristic
60C2×C22⋊C8C2×C412yes—
61C2×#97—22no—
62C2×C4×C8—22no—
63#381—21no—
64#381—21no—
65#381—21no—
66#381—21no—
67#381—21no—
68#380—21no—
69#380—21no—
70#379—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4C2×C412yes—
75C4×C22⋊C4D411yescharacteristic
76C4×C22⋊C4D411yescharacteristic
77C4×C22⋊C4D412yes—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95C2×C412yes—
81#336—22no—
82#336—22no—
83C25#4111yescharacteristic
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—82no—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×Q811yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—82no—
118#117—82no—
119C2×C4⋊C4—41no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—82no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—82no—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C412yes—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8C4212yes—
151C22⋊C8C4212yes—
152C22⋊C8C4⋊C414yes—
153#97—82no—
154C4×C8—82no—
155#95—21no—
156#95—22no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164C24—42no—
165C24—42no—
166C24—42no—
167C24#11711yescharacteristic
168C24#11711yescharacteristic
169C24#11711yescharacteristic
170C24#12311yescharacteristic
171C24#12311yescharacteristic
172C24#12411yescharacteristic
173C24#12511yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4C2×C4211yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C4⋊C411yescharacteristic
201C22×C4#11711yescharacteristic
202C22×C4#11711yescharacteristic
203C22×C4#11711yescharacteristic
204C22×C4#11711yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×Q811yescharacteristic
213C22×C4C4×Q811yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12611yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226M16—82no—
227M16—82no—
228M16—82no—
229M16—82no—
230M16—82no—
231M16—82no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—82no—
246C22⋊C4—24no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—42no—
255C42—42no—
256C42—82no—
257C42—82no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—88no—
280C23#37611yescharacteristic
281C23#38111yescharacteristic
282C23#38211yescharacteristic
283C23#38711yescharacteristic
284C23#38711yescharacteristic
285C23#38811yescharacteristic
286C23#38811yescharacteristic
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—88no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—164no—
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4C4×C4⋊C411yescharacteristic
331C2×C4#38011yescharacteristic
332C2×C4#38311yescharacteristic
333C2×C4#38711yescharacteristic
334C2×C4#38711yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C8—42no—
338C8—42no—
339C8—44no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#143411yescharacteristic
368C22#174712yes—
369C22#182812yes—
370C22#193212yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—82no—
381C4—162no—
382C4—162no—
383C2—22no—
384C2—22no—
385C2—22no—
386C2—41no—
387C2—41no—
388C2—41no—
389C2—41no—
390C2#1079911yescharacteristic
391C2#1082611yescharacteristic
392C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1749C4semidirect
19. C4×C22⋊C8C4semidirect
25. #1282C4semidirect
26. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#205252
#80092
#79432
#79422
C2×#12842
#79252
#79132