Finite Group Databasebeta

← #203078

Subgroups — #203078

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17823C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17491C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#17325C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8#8066C211yesmaximal, characteristic
9#7886C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#1574C2214yes—
18#1572—22no—
19#1569—22no—
20#1552C414yes—
21#1549C412yes—
22#1546C412yes—
23#1544C2212yes—
24#1544C2212yes—
25#1497—22no—
26#1497—21no—
27#1497—21no—
28#1493C2214yes—
29#1487—22no—
30#1485C2214yes—
31#1479—22no—
32#1453—22no—
33#1452—21no—
34#1452C2212yes—
35#1450—21no—
36#1450C2212yes—
37#1299C2212yes—
38#1299C2212yes—
39#1278C2214yes—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533C2312yes—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432D412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431C2×C412yes—
76#431D411yescharacteristic
77#431D411yescharacteristic
78#431D412yes—
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81#431C2312yes—
82#431C2312yes—
83#423—21no—
84#423C2×C412yes—
85#423D412yes—
86#423C2312yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89#407—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16C2312yes—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#335—24no—
102#334—24no—
103#332C2×C414yes—
104#332C2×C414yes—
105#332C2×C414yes—
106#332C2×C414yes—
107Q8⋊C8—28no—
108Q8⋊C8—24no—
109Q8⋊C8—24no—
110#322—22no—
111#322—22no—
112#322—22no—
113C82—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C22×C412yes—
132C4⋊Q8C22×C412yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—22no—
154#117—22no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22×C412yes—
188C4⋊C8C22×C412yes—
189C4⋊C8C22×C414yes—
190C4⋊C8C2×D414yes—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205#97—22no—
206#97—22no—
207#97—22no—
208#97—41no—
209C4×C8—22no—
210C4×C8—21no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—41no—
217C4×C8C2×D412yes—
218C4×C8C2×D412yes—
219C24#12611yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—82no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4#11711yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249M16—44no—
250M16—82no—
251M16—82no—
252M16—82no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—42no—
262C2×C8—42no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C42—42no—
278C42—42no—
279C42#11711yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42#12311yescharacteristic
288C42#12311yescharacteristic
289C42#12411yescharacteristic
290C42#12411yescharacteristic
291C42C23×C411yescharacteristic
292C42C22×D411yescharacteristic
293C42C22×D411yescharacteristic
294C42C2×#4111yescharacteristic
295C23—41no—
296C23—41no—
297C23—41no—
298C23#38811yescharacteristic
299C23#38811yescharacteristic
300C23C2×#12611yescharacteristic
301Q8—44no—
302Q8—44no—
303Q8—82no—
304Q8—82no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
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—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4C2×#11711yescharacteristic
334C2×C4C2×C4×D411yescharacteristic
335C2×C4C2×C4×D411yescharacteristic
336C2×C4C2×#12111yescharacteristic
337C2×C4C2×#12311yescharacteristic
338C2×C4C2×#12411yescharacteristic
339C8—42no—
340C8—42no—
341C8—42no—
342C8—42no—
343C8—44no—
344C8—44no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—82no—
357C22C2×#38811yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—82no—
369C4#295512yes—
370C4#295712yes—
371C4#306912yes—
372C2—41no—
373C2—41no—
374C2—41no—
375C2#2107011yescharacteristic
376C2#2107711yescharacteristic
377C2#2109811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#126
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #9587C2semidirect
7. #9585C2semidirect

Maximal subgroups

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

subgroupindex
#178232
#177152
#174912
#174792
#173252
#95872
#95852
#80662
#78862