Finite Group Databasebeta

← #117162

Subgroups — #117162

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#20801C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—22no—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#382C2211yescharacteristic
16#1998—22no—
17#1997C412yescomplemented
18#1997C412yescomplemented
19#1847—22no—
20#1846—22no—
21#1742—22no—
22#1520—22no—
23#1518—22no—
24#1518—22no—
25#1518—22no—
26#1503—22no—
27C2×#328—22no—
28#1303—22no—
29#1286—22no—
30#1278—22no—
31C22×M16—21no—
32C22×M16D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8Q812yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431D412yes—
59#431D412yes—
60#428—44no—
61#428—44no—
62C2×#107—24no—
63C2×#106—24no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C4×M16D412yes—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#384—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#382D412yes—
87#382D412yes—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C2×#95—21no—
93C2×#95—21no—
94#343—24no—
95#343—24no—
96#328—28no—
97#328—28no—
98C22×Q8—21no—
99C22×Q8C4⋊C412yes—
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16C22⋊C414yes—
112C2×M16SD1614yes—
113C2×M16Q1614yes—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C4⋊C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—41no—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181#107—48no—
182#106—48no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209C24#12111yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—82no—
215C2×Q8—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C4⋊C411yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12711yescharacteristic
243C22×C4C2×SD1612yes—
244C22×C4C2×Q1612yes—
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4#13712yes—
247M16—24no—
248M16—24no—
249M16—24no—
250M16—44no—
251M16—81no—
252M16—81no—
253M16#10318yes—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—82no—
285C42—42no—
286C42—42no—
287C42—81no—
288C42—81no—
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C42#12411yescharacteristic
304C42#12411yescharacteristic
305C23—41no—
306C23—41no—
307C23#38011yescharacteristic
308C23#39211yescharacteristic
309C23#39711yescharacteristic
310C23#46012yes—
311C23#46212yes—
312Q8—82no—
313Q8—82no—
314Q8—82no—
315Q8—82no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—164no—
339C2×C4#38511yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#39011yescharacteristic
345C2×C4#39311yescharacteristic
346C2×C4#39411yescharacteristic
347C2×C4#39411yescharacteristic
348C2×C4#39611yescharacteristic
349C2×C4#39611yescharacteristic
350C2×C4#39911yescharacteristic
351C2×C4C2×#10312yes—
352C2×C4#41712yes—
353C2×C4#45012yes—
354C2×C4#45112yes—
355C2×C4#47512yes—
356C2×C4#47712yes—
357C2×C4#48312yes—
358C2×C4#48412yes—
359C2×C4#48712yes—
360C2×C4#48912yes—
361C2×C4#49211yescharacteristic
362C2×C4#49411yescharacteristic
363C2×C4#49611yescharacteristic
364C2×C4#49711yescharacteristic
365C8—28no—
366C8—82no—
367C8—82no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—42no—
376C22—42no—
377C22#143911yescharacteristic
378C22#180312yes—
379C22#201411yescharacteristic
380C22#201611yescharacteristic
381C22#207212yes—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—82no—
392C4—162no—
393C4—162no—
394C4#187212yes—
395C4#195611yescharacteristic
396C4#195911yescharacteristic
397C4#202412yes—
398C4#209511yescharacteristic
399C4#209611yescharacteristic
400C2—21no—
401C2—22no—
402C2—22no—
403C2—21no—
404C2#1105011yescharacteristic
405C2#1113811yescharacteristic
406C2#1114111yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1997C4semidirect
18. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#213372
#208012
#175752
C2×#15182
#80952
C2×#12862
#78822