Finite Group Databasebeta

← #107553

Subgroups — #107553

Order 512.

701 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#17672C211yesmaximal, characteristic
3C2×#1531C211yesmaximal, characteristic
4#8125C211yesmaximal, characteristic
5C2×#1302C211yesmaximal, characteristic
6#8080C211yesmaximal, characteristic
7#7919C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983—21no—
13#2982—21no—
14#2962—22no—
15#2960—22no—
16#2924C414yes—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#1537C414yes—
24#1531—22no—
25#1531—22no—
26#1531C414yes—
27#1528—22no—
28C2×#330—22no—
29#1308—22no—
30#1302—21no—
31#1302—21no—
32#1301—22no—
33#1281—22no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533D412yes—
43C2×C4⋊Q8Q811yescharacteristic
44C2×C4⋊Q8Q811yescharacteristic
45C2×#125D412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117C2×C412yes—
51C22×C42C2311yescharacteristic
52#446—24no—
53#446—24no—
54#446—24no—
55#446—28no—
56C8×Q8—24no—
57C8×Q8—24no—
58C8×Q8—24no—
59#433—28no—
60#432—21no—
61#432—24no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432C2×C412yes—
70#432C2×C412yes—
71#432C2×C412yes—
72#432D412yes—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—24no—
77#431—21no—
78#431—21no—
79#431D412yes—
80#431D412yes—
81#431D411yescharacteristic
82#431D411yescharacteristic
83#424—24no—
84#423—21no—
85#423—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89#407—21no—
90#407C2×C412yes—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8C2×C412yes—
94#405—24no—
95C4×M16—21no—
96C4×M16—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97C2×C412yes—
105C2×C4×C8—21no—
106#330—28no—
107#330—28no—
108C22×Q8—21no—
109C22×Q8—21no—
110C23×C4C22×C411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—44no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129C4⋊Q8C4⋊C412yes—
130#125—22no—
131#125—22no—
132#125C22⋊C414yes—
133#125C22⋊C414yes—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—22no—
151#117—22no—
152#117C4214yes—
153#117C22⋊C414yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C4⋊C412yes—
159C2×C4⋊C4C4⋊C412yes—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—28no—
175C4⋊C8—28no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8C22⋊C412yes—
196C22⋊C8C22⋊C412yes—
197C22⋊C8C22⋊C414yes—
198#97—28no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—41no—
206#97—41no—
207#97—41no—
208#97—41no—
209C4×C8—28no—
210C4×C8—21no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8—41no—
215C24#11711yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8—82no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245M16—44no—
246M16—82no—
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4#9514yes—
272C4⋊C4#9514yes—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C42—24no—
276C42—24no—
277C42C2×C22⋊C411yescharacteristic
278C42C2×C22⋊C411yescharacteristic
279C42C2×C4⋊C411yescharacteristic
280C42C2×C4⋊C411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42#12011yescharacteristic
286C42#12011yescharacteristic
287C42#12011yescharacteristic
288C42#12011yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C23—22no—
294C23—22no—
295C23—41no—
296C23C4×C22⋊C411yescharacteristic
297C23#38011yescharacteristic
298C23#38111yescharacteristic
299Q8—44no—
300Q8—44no—
301Q8—44no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4C2×#9511yescharacteristic
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4#37911yescharacteristic
328C2×C4#38011yescharacteristic
329C2×C4#38511yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38611yescharacteristic
337C8—42no—
338C8—42no—
339C8—44no—
340C8—44no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22C22⋊#9511yescharacteristic
353C22#181412yes—
354C22#183012yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—44no—
367C4—44no—
368C4#186112yes—
369C2—22no—
370C2—22no—
371C2—41no—
372C2#1077011yescharacteristic
373C2#1077311yescharacteristic
374C2#1078011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#602502
#176722
C2×#15312
#81252
C2×#13022
#80802
#79192