Finite Group Databasebeta

← #118034

Subgroups — #118034

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17806C211yesmaximal, characteristic
2#17562C211yesmaximal, characteristic
3#17486C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8061C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2283C2211yescharacteristic
15#1574—22no—
16#1573—22no—
17#1564—22no—
18#1515—22no—
19#1511—22no—
20#1503—22no—
21#1493—22no—
22#1487C414yescomplemented
23#1481—22no—
24#1472—22no—
25#1470—22no—
26#1461—22no—
27#1299—22no—
28#1279—22no—
29#1274—22no—
30C22×M16—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8D412yes—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×#117—21no—
47C22×C42C2311yescharacteristic
48#432—21no—
49#432—21no—
50#432—21no—
51#432D412yes—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431D412yes—
62#431D412yes—
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⋊C8—21no—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C2×C4×C8—21no—
81C2×C4×C8D412yes—
82#383—21no—
83#382—21no—
84#381—21no—
85#380—21no—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—21no—
88#376—21no—
89#335—24no—
90#334—24no—
91#328—28no—
92Q8⋊C8—28no—
93#324—28no—
94C22×Q8—21no—
95C22×Q8C4⋊C412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—82no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8—41no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115#125—41no—
116#125—41no—
117#125—41no—
118#125—41no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—41no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C4⋊C412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C4⋊C8—22no—
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—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182#97—22no—
183#97—41no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8C22⋊C414yes—
190#95—22no—
191#95—41no—
192#95—41no—
193#95—41no—
194C24#12311yescharacteristic
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226M16—82no—
227M16—82no—
228M16—82no—
229M16—82no—
230C2×C8—24no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—82no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42C2×C22⋊C411yescharacteristic
259C42#11711yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42#12011yescharacteristic
263C42#12111yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42#12311yescharacteristic
272C42#12411yescharacteristic
273C42#12711yescharacteristic
274C23—22no—
275C23—41no—
276C23—41no—
277C23#38511yescharacteristic
278C23#39411yescharacteristic
279C23#39711yescharacteristic
280Q8—44no—
281Q8—82no—
282Q8—82no—
283Q8—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
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—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4#38011yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#39011yescharacteristic
316C2×C4#39211yescharacteristic
317C2×C4#39311yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39911yescharacteristic
322C2×C4#47112yes—
323C2×C4#47312yes—
324C8—44no—
325C8—82no—
326C8—82no—
327C8—82no—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—82no—
338C22#143911yescharacteristic
339C22#205212yes—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—82no—
350C4—82no—
351C4—162no—
352C4#197711yescharacteristic
353C4#197911yescharacteristic
354C4#206412yes—
355C4#209812yes—
356C2—22no—
357C2—41no—
358C2—41no—
359C2#1108411yescharacteristic
360C2#1110211yescharacteristic
361C2#1112811yescharacteristic
subgroupG/H
centerC2×C4#390
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
22. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#178062
#175622
#174862
#173852
#80612
#78992
#78582