Finite Group Databasebeta

← #118776

Subgroups — #118776

Order 512.

513 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17751C211yesmaximal, characteristic
2#17575C211yesmaximal, characteristic
3#17496C211yesmaximal, characteristic
4#8109C211yesmaximal, characteristic
5#8065C211yesmaximal, characteristic
6#8042C211yesmaximal, characteristic
7#7863C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2285C2211yescharacteristic
15#1561—22no—
16#1552—22no—
17#1549C412yescomplemented
18#1546C412yescomplemented
19#1520—22no—
20#1518—22no—
21#1503—22no—
22#1497—22no—
23#1491—22no—
24#1481—22no—
25#1305—22no—
26#1299—22no—
27#1296—22no—
28#1275—22no—
29C22×M16—21no—
30#533—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533D412yes—
37#533D412yes—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×#125—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×#117—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#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16D412yes—
80C2×#97—21no—
81C2×C4×C8—21no—
82#384—21no—
83#382—21no—
84#380—21no—
85#380D412yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#332—24no—
89#332—24no—
90#328—28no—
91Q8⋊C8—28no—
92C22×Q8—21no—
93C22×Q8—21no—
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8—41no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8C22⋊C412yes—
112C4⋊Q8C22⋊C412yes—
113#125—22no—
114#125—22no—
115#125—41no—
116#125—41no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133#117—22no—
134#117—22no—
135#117—41no—
136#117—41no—
137#117—82no—
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×C4⋊C4—41no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#97—22no—
184#97—41no—
185#97—41no—
186#97—41no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—41no—
191#95—22no—
192#95—41no—
193C24#12311yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4#11711yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12411yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227M16—82no—
228M16—82no—
229M16—82no—
230M16—82no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—82no—
259C42—42no—
260C42—42no—
261C42—82no—
262C42C2×C22⋊C411yescharacteristic
263C42#11711yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42#12011yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12311yescharacteristic
271C42#12311yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42#12311yescharacteristic
275C42#12311yescharacteristic
276C42#12411yescharacteristic
277C42#12611yescharacteristic
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23#38811yescharacteristic
283C23#39411yescharacteristic
284C23#39711yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
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—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4#38111yescharacteristic
319C2×C4#38611yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#39411yescharacteristic
325C2×C4#39411yescharacteristic
326C2×C4#39411yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#39711yescharacteristic
330C2×C4#47112yes—
331C2×C4#47312yes—
332C8—82no—
333C8—82no—
334C8—82no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—81no—
346C22—81no—
347C22#144011yescharacteristic
348C22#204611yescharacteristic
349C22#204711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—82no—
361C4—162no—
362C4#190211yescharacteristic
363C4#190311yescharacteristic
364C4#193012yes—
365C4#205212yes—
366C2—21no—
367C2—21no—
368C2—41no—
369C2—41no—
370C2#1116411yescharacteristic
371C2#1118011yescharacteristic
372C2#1118511yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
17. #1549C4semidirect
18. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#177512
#175752
#174962
#81092
#80652
#80422
#78632