Finite Group Databasebeta

← #117770

Subgroups — #117770

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21822C211yesmaximal, characteristic
2#17797C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4C2×#1491C211yesmaximal, characteristic
5#8085C211yesmaximal, characteristic
6#8056C211yesmaximal, characteristic
7#7994C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1985—22no—
16#1985—22no—
17#1817—22no—
18#1569—22no—
19#1568—22no—
20#1565—22no—
21#1491—22no—
22#1491—22no—
23#1491C412yes—
24#1491C412yes—
25#1487—22no—
26#1479—22no—
27C2×Q8⋊C8—22no—
28#1302—22no—
29#1298—22no—
30#1291—22no—
31C23×Q8Q811yescharacteristic
32C22×M16—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4Q811yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431D412yes—
69#423—21no—
70#423D412yes—
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—
78C2×C22⋊C8D412yes—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16C2×C411yescharacteristic
84C4×M16C2×C411yescharacteristic
85C4×M16D412yes—
86C2×#97—21no—
87C2×#97—21no—
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8D412yes—
91#336—28no—
92#335—24no—
93#334—24no—
94Q8⋊C8—28no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—44no—
103C22×Q8C4⋊C412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×Q811yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—44no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C4⋊C412yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—21no—
186#97—21no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191#97—41no—
192#97—41no—
193#97—41no—
194#97—41no—
195#97—41no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8C22⋊C412yes—
200C4×C8C22⋊C412yes—
201C24C22⋊Q811yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—82no—
226C22×C4C2×C4⋊C411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12711yescharacteristic
244M16—24no—
245M16—44no—
246M16—42no—
247M16—82no—
248M16—82no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42C2×C22⋊C411yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12011yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12311yescharacteristic
284C42#12411yescharacteristic
285C42#12411yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C23—22no—
289C23—22no—
290C23—41no—
291C23#38511yescharacteristic
292C23#39611yescharacteristic
293C23#39911yescharacteristic
294Q8—24no—
295Q8—24no—
296Q8—42no—
297Q8—42no—
298Q8—44no—
299Q8—44no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#38011yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#39211yescharacteristic
334C2×C4#39311yescharacteristic
335C2×C4#39411yescharacteristic
336C2×C4#39411yescharacteristic
337C2×C4#39611yescharacteristic
338C2×C4#39711yescharacteristic
339C8—24no—
340C8—42no—
341C8—82no—
342C8—82no—
343C8—82no—
344C8—82no—
345C8—82no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—44no—
354C22#143911yescharacteristic
355C22#186112yes—
356C22#209112yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—44no—
369C4—82no—
370C4#200912yes—
371C2—22no—
372C2—22no—
373C2—41no—
374C2#1108011yescharacteristic
375C2#1110111yescharacteristic
376C2#1113011yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
LCSG ▷ C2×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
#218222
#177972
#174492
C2×#14912
#80852
#80562
#79942