Finite Group Databasebeta

← #117817

Subgroups — #117817

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2#21839C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4C2×#1485C211yesmaximal, characteristic
5C2×#1291C211yesmaximal, characteristic
6#7914C211yesmaximal, characteristic
7#7900C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993—21no—
16#1992—22no—
17#1988—21no—
18#1987—21no—
19#1960—21no—
20#1872—22no—
21#1836—22no—
22#1835—22no—
23#1825—22no—
24#1805—22no—
25#1736—22no—
26#1485—22no—
27#1485C414yescomplemented
28C2×Q8⋊C8—22no—
29#1291—22no—
30#1280—22no—
31#1279—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533Q812yes—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4D412yes—
51C22×C42C2311yescharacteristic
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#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×#103—24no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8D412yes—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16C2×C412yes—
74C4×M16D412yes—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8D412yes—
80#384D412yes—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C2×#95—21no—
91C2×#95—21no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—28no—
96Q8⋊C8—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—42no—
106C2×M16—42no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4⋊C412yes—
118C4⋊Q8C4⋊C412yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—41no—
142#117—41no—
143#117—44no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—44no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178#103—84no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—22no—
187#97—41no—
188#97—41no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8C22⋊C414yes—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201C24#12111yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—44no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—42no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#11711yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12711yescharacteristic
238M16—44no—
239M16—44no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—42no—
271C42—42no—
272C42—82no—
273C42C2×C22⋊C411yescharacteristic
274C42C2×C4⋊C411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12311yescharacteristic
286C42#12311yescharacteristic
287C42#12411yescharacteristic
288C42#12411yescharacteristic
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23#38811yescharacteristic
294C23#39211yescharacteristic
295C23#39611yescharacteristic
296Q8—44no—
297Q8—44no—
298Q8—82no—
299Q8—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
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—42no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—164no—
330C2×C4#38011yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39311yescharacteristic
337C2×C4#39411yescharacteristic
338C2×C4#39411yescharacteristic
339C2×C4#39611yescharacteristic
340C2×C4#39711yescharacteristic
341C2×C4#39911yescharacteristic
342C2×C4#45412yes—
343C2×C4#45512yes—
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22#143911yescharacteristic
357C22#200311yescharacteristic
358C22#200411yescharacteristic
359C22#200912yes—
360C22#201712yes—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4—162no—
374C4—162no—
375C4#186012yes—
376C4#204012yes—
377C2—41no—
378C2—41no—
379C2#1012312yes—
380C2#1012512yes—
381C2#1107911yescharacteristic
382C2#1111111yescharacteristic
383C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#218392
#207652
C2×#14852
C2×#12912
#79142
#79002