Finite Group Databasebeta

← #108071

Subgroups — #108071

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2985C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic
3#17715C211yesmaximal, characteristic
4#8090C211yesmaximal, characteristic
5#7948C211yesmaximal, characteristic
6#7886C211yesmaximal, characteristic
7#7854C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2985—22no—
10#2985—22no—
11#2985—22no—
12#2985C414yescomplemented
13C2×#446—22no—
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#433—22no—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1994C414yes—
24#1986—22no—
25#1826—22no—
26#1552—22no—
27#1549C412yescomplemented
28#1546C412yescomplemented
29#1544—22no—
30#1302—22no—
31#1285—22no—
32#1278—21no—
33#1278—21no—
34#1274—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533D412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125Q812yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117D412yes—
52C22×C42C2311yescharacteristic
53#446—28no—
54#446—24no—
55#446—28no—
56#446—24no—
57C8×Q8—24no—
58C8×Q8—24no—
59#433—28no—
60#433—28no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432C2×C412yes—
68#432D412yes—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431C2×C412yes—
79#431C2×C412yes—
80#431D412yes—
81#423C2×C412yes—
82#423C2×C412yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#407—21no—
89#407—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C4×M16—21no—
94C4×M16—21no—
95C2×#97—22no—
96C2×#97D412yes—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105#376—21no—
106C43C2×C412yes—
107#339—28no—
108#332—24no—
109#332—24no—
110C22×Q8—21no—
111C22×Q8—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—22no—
116C2×M16—22no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4212yes—
126C4⋊Q8C4212yes—
127C4⋊Q8C22⋊C412yes—
128C4⋊Q8C22⋊C412yes—
129#125—22no—
130#125—22no—
131#125C4⋊C414yes—
132#125C4⋊C414yes—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—22no—
150#117—22no—
151#117—24no—
152#117—42no—
153#117C22⋊C414yes—
154#117C22⋊C414yes—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C4⋊C412yes—
161C2×C4⋊C4C4⋊C412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—28no—
181C4⋊C8—28no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202#97—28no—
203#97—22no—
204#97—41no—
205#97—41no—
206#97C22⋊C414yes—
207C4×C8—28no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8C22⋊C412yes—
215C4×C8C22⋊C412yes—
216#95—41no—
217#95—41no—
218C24#12311yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251M16—44no—
252M16—42no—
253M16—42no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—82no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—44no—
275C4⋊C4#9514yes—
276C4⋊C4#9514yes—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—44no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—42no—
284C42—44no—
285C42—82no—
286C42C2×C4211yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42C2×C22⋊C411yescharacteristic
289C42C2×C22⋊C411yescharacteristic
290C42C2×C4⋊C411yescharacteristic
291C42C2×C4⋊C411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42#12011yescharacteristic
297C42#12011yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C23—22no—
303C23—41no—
304C23—41no—
305C23#38111yescharacteristic
306C23#38511yescharacteristic
307C23#38611yescharacteristic
308Q8—44no—
309Q8—42no—
310Q8—42no—
311Q8—82no—
312Q8—82no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4C2×#9511yescharacteristic
340C2×C4C4×C22⋊C411yescharacteristic
341C2×C4C4×C22⋊C411yescharacteristic
342C2×C4#37911yescharacteristic
343C2×C4#38011yescharacteristic
344C2×C4#38011yescharacteristic
345C2×C4#38511yescharacteristic
346C2×C4#38511yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38611yescharacteristic
349C2×C4#38611yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#44312yes—
352C2×C4#44412yes—
353C8—42no—
354C8—44no—
355C8—42no—
356C8—82no—
357C8—82no—
358C8—82no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—82no—
368C22C22⋊#9511yescharacteristic
369C22#186612yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—44no—
381C4—84no—
382C4#176411yescharacteristic
383C4#176511yescharacteristic
384C4#187612yes—
385C4#191212yes—
386C2—22no—
387C2—41no—
388C2—41no—
389C2#1074111yescharacteristic
390C2#1078711yescharacteristic
391C2#1079111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#123
LCSG ▷ 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
12. #2985C4semidirect
27. #1549C4semidirect
28. #1546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
#218442
#177152
#80902
#79482
#78862
#78542