Finite Group Databasebeta

← #110764

Subgroups — #110764

Order 512.

623 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#17718C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3C2×#1493C211yesmaximal, characteristic
4C2×C8×M16C211yesmaximal, characteristic
5#8081C211yesmaximal, characteristic
6#8063C211yesmaximal, characteristic
7#7885C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#1554—22no—
16#1548C412yes—
17#1546C412yes—
18#1542—22no—
19#1531C414yes—
20#1528—22no—
21#1526—22no—
22#1493—21no—
23#1493—21no—
24#1493C414yes—
25C2×Q8⋊C8—22no—
26C8×M16—22no—
27C8×M16—22no—
28C2×#322—21no—
29C2×C82—21no—
30#1301—22no—
31#1299—22no—
32#1278—22no—
33C22×M16—21no—
34C22×M16—21no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16D412yes—
82C4×M16D412yes—
83C2×#97C2×C412yes—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C412yes—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D412yes—
93#333—24no—
94#332—24no—
95#330—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98#322—24no—
99C82—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—28no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—82no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—44no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
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×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—22no—
190#97—22no—
191#97—41no—
192#97C22⋊C414yes—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—21no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—21no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8C22⋊C412yes—
204C4×C8C22⋊C412yes—
205C24#12611yescharacteristic
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—44no—
211C2×Q8—82no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4#11711yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4#11711yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236M16—28no—
237M16—28no—
238M16—44no—
239M16—82no—
240M16—82no—
241M16—82no—
242C2×C8—24no—
243C2×C8—28no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—82no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—42no—
272C42—42no—
273C42C2×C4211yescharacteristic
274C42C2×C22⋊C411yescharacteristic
275C42C2×C22⋊C411yescharacteristic
276C42C2×C22⋊C411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42#12011yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12611yescharacteristic
288C42#12711yescharacteristic
289C23—22no—
290C23—22no—
291C23—41no—
292C23#38811yescharacteristic
293C23#38811yescharacteristic
294C23#38811yescharacteristic
295Q8—44no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4C4×C22⋊C411yescharacteristic
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4C4×C22⋊C411yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38611yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#39011yescharacteristic
331C2×C4#39011yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#44314yes—
334C2×C4#47112yes—
335C2×C4#47312yes—
336C8—24no—
337C8—28no—
338C8—42no—
339C8—44no—
340C8—82no—
341C8—82no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#143711yescharacteristic
352C22#193012yes—
353C22#193712yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—82no—
363C4#176412yes—
364C4#190812yes—
365C4#193212yes—
366C4#197412yes—
367C4#197711yescharacteristic
368C4#197911yescharacteristic
369C2—22no—
370C2—22no—
371C2—41no—
372C2#1092911yescharacteristic
373C2#1094211yescharacteristic
374C2#1094611yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#126
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#177182
#176482
C2×#14932
C2×C8×M162
#80812
#80632
#78852