Finite Group Databasebeta

← #117552

Subgroups — #117552

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#21302C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17715C211yesmaximal, characteristic
5#8084C211yesmaximal, characteristic
6#8068C211yesmaximal, characteristic
7#7891C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—21no—
16#1994—21no—
17#1986—22no—
18#1985—22no—
19#1984—22no—
20#1826—22no—
21#1729—22no—
22#1573—22no—
23#1571—21no—
24#1570—21no—
25#1565—22no—
26#1552—22no—
27#1549C412yescomplemented
28#1546C412yescomplemented
29#1544—22no—
30#1302—22no—
31#1300—22no—
32#1279—22no—
33C22×M16—21no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125Q812yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117D412yes—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432D412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65#423C2×C412yes—
66#423D411yescharacteristic
67#423D411yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75#407—21no—
76#407—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×#97D412yes—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92C43—21no—
93#339—24no—
94#339—24no—
95#336—28no—
96#335—24no—
97#334—24no—
98#332—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—24no—
106C2×M16—22no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—42no—
110C2×M16—41no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—44no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—24no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C4⋊C412yes—
158C2×C4⋊C4C4⋊C412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8C22⋊C414yes—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—41no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—41no—
204C4×C8—41no—
205#95—22no—
206#95—41no—
207C24#12311yescharacteristic
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—82no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12611yescharacteristic
241M16—44no—
242M16—44no—
243M16—42no—
244M16—42no—
245M16—42no—
246M16—42no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—82no—
282C42C2×C22⋊C411yescharacteristic
283C42#11711yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12011yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42#12311yescharacteristic
296C42#12411yescharacteristic
297C42#12711yescharacteristic
298C23—22no—
299C23—22no—
300C23—41no—
301C23#38511yescharacteristic
302C23#39411yescharacteristic
303C23#39711yescharacteristic
304Q8—82no—
305Q8—82no—
306Q8—82no—
307Q8—81no—
308Q8—81no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#38011yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#39011yescharacteristic
342C2×C4#39211yescharacteristic
343C2×C4#39311yescharacteristic
344C2×C4#39411yescharacteristic
345C2×C4#39611yescharacteristic
346C2×C4#39611yescharacteristic
347C2×C4#39911yescharacteristic
348C2×C4#44312yes—
349C2×C4#44412yes—
350C2×C4#45412yes—
351C2×C4#45612yes—
352C8—44no—
353C8—44no—
354C8—82no—
355C8—82no—
356C8—81no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—44no—
366C22#143911yescharacteristic
367C22#186612yes—
368C22#203812yes—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—84no—
378C4—82no—
379C4—82no—
380C4#185712yes—
381C4#187612yes—
382C4#193711yescharacteristic
383C4#193811yescharacteristic
384C4#200311yescharacteristic
385C4#200511yescharacteristic
386C2—22no—
387C2—22no—
388C2—41no—
389C2#1106811yescharacteristic
390C2#1113111yescharacteristic
391C2#1113611yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1549C4semidirect
28. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#213022
#178072
#177152
#80842
#80682
#78912