Finite Group Databasebeta

← #116144

Subgroups — #116144

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21352C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17590C211yesmaximal, characteristic
5#8074C211yesmaximal, characteristic
6#7998C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1940—22no—
16#1851—22no—
17#1768—22no—
18#1531—22no—
19#1528—22no—
20#1526—22no—
21#1520—22no—
22#1518C414yescomplemented
23#1515—22no—
24#1511—22no—
25#1506—22no—
26#1506—22no—
27#1300—22no—
28#1291—22no—
29#1284—22no—
30C22×M16—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533D412yes—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—21no—
47C22×C42C2311yescharacteristic
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432D412yes—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#423—21no—
61#423—21no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—21no—
64#407—21no—
65#407—21no—
66C2×C22⋊C8D412yes—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C2×#97—21no—
74C2×#97D412yes—
75C2×C4×C8—21no—
76#383—21no—
77#383—21no—
78#382—21no—
79#382C2×C412yes—
80#381—21no—
81#381D412yes—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88#376D412yes—
89#376D412yes—
90#336—24no—
91#330—28no—
92#328—28no—
93#328—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—44no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C4⋊C412yes—
113C4⋊Q8C4⋊C412yes—
114#125—22no—
115#125—22no—
116#125—41no—
117#125—41no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—22no—
135#117—22no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—48no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—41no—
181#97—41no—
182#97—41no—
183C4×C8—22no—
184C4×C8—41no—
185C4×C8—41no—
186C4×C8—41no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—41no—
193#95—41no—
194#95C22⋊C414yes—
195C24#12111yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12611yescharacteristic
227M16—44no—
228M16—44no—
229M16—82no—
230M16—82no—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C42—44no—
257C42—42no—
258C42—42no—
259C42—82no—
260C42C2×C4⋊C411yescharacteristic
261C42#11711yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42#12011yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42#12311yescharacteristic
271C42#12311yescharacteristic
272C42#12411yescharacteristic
273C42#12411yescharacteristic
274C42#12611yescharacteristic
275C42#12711yescharacteristic
276C23—22no—
277C23—41no—
278C23—41no—
279C23#38611yescharacteristic
280C23#39411yescharacteristic
281C23#39611yescharacteristic
282Q8—82no—
283Q8—82no—
284Q8—82no—
285Q8—82no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
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—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#38011yescharacteristic
315C2×C4#38511yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#39011yescharacteristic
320C2×C4#39211yescharacteristic
321C2×C4#39311yescharacteristic
322C2×C4#39411yescharacteristic
323C2×C4#39611yescharacteristic
324C2×C4#39711yescharacteristic
325C2×C4#39911yescharacteristic
326C8—44no—
327C8—82no—
328C8—82no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—82no—
339C22#143911yescharacteristic
340C22#190212yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—84no—
353C4—82no—
354C4—82no—
355C4—162no—
356C4#203112yes—
357C2—22no—
358C2—41no—
359C2—41no—
360C2#1105911yescharacteristic
361C2#1108511yescharacteristic
362C2#1109411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#213522
#176482
#175962
#175902
#80742
#79982
#79442