Finite Group Databasebeta

← #102297

Subgroups — #102297

Order 512.

542 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2C8×#95C211yesmaximal, characteristic
3#9909C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic
5#8595C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7665C211yesmaximal, characteristic
8C2×#487C411yescharacteristic, complemented
9C22×C4×C8—21no—
10C22×C4×C8C2211yescharacteristic
11#2857C2211yescharacteristic
12#2857C2211yescharacteristic
13#1990C411yescharacteristic, complemented
14#1956C411yescharacteristic, complemented
15#1933—21no—
16#1933—21no—
17#1892—21no—
18#1892—21no—
19#1834C411yescharacteristic, complemented
20#1797C2211yescharacteristic
21#1797C2211yescharacteristic
22C2×#336—21no—
23C2×#336C2211yescharacteristic
24C2×#334—21no—
25C2×#328—22no—
26C4×#95C2211yescharacteristic
27#1294—22no—
28#1277—21no—
29#1271—22no—
30#1271—22no—
31C23×C8—21no—
32C23×C8D411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#487—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C4×C8—24no—
50C2×C4×C8—42no—
51C2×C4×C8—42no—
52C2×C4×C8—42no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57#391—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#384C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65#380—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69C2×#95—21no—
70C2×#95D412yes—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—24no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#334—42no—
81#328—44no—
82C22×Q8—21no—
83C22×Q8—21no—
84C23×C4—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C4⋊Q8—41no—
110C4⋊Q8—41no—
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C22⋊Q8—42no—
114C22⋊Q8—41no—
115C22⋊Q8—41no—
116C22⋊Q8—42no—
117C22⋊Q8—41no—
118C22⋊Q8—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149#103—42no—
150#103—42no—
151#103—42no—
152#103—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—48no—
160#95—24no—
161#95—22no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95SD1614yes—
167C24#12111yescharacteristic
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—81no—
176C2×Q8—81no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12411yescharacteristic
202C22×C4#12711yescharacteristic
203C22×C4C2×SD1612yes—
204C22×C4#13511yescharacteristic
205C22×C4#13511yescharacteristic
206C22×C4#13512yes—
207C22×C4AGL(1,ℤ/8)11yescharacteristic
208C22×C4#13711yescharacteristic
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—24no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C2×C8#9912yes—
236C2×C8#10412yes—
237C2×C8#10412yes—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#39011yescharacteristic
273C23#46311yescharacteristic
274C23#46411yescharacteristic
275C23#46512yes—
276C23#47312yes—
277Q8—81no—
278Q8—81no—
279Q8—81no—
280Q8—81no—
281Q8—162no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×#9911yescharacteristic
324C2×C4#41011yescharacteristic
325C2×C4#41611yescharacteristic
326C2×C4#41711yescharacteristic
327C2×C4C2×#10411yescharacteristic
328C2×C4C2×#10411yescharacteristic
329C2×C4C4×SD1612yes—
330C2×C4#44312yes—
331C2×C4#46311yescharacteristic
332C2×C4#46411yescharacteristic
333C2×C4#49211yescharacteristic
334C2×C4#49511yescharacteristic
335C8—22no—
336C8—22no—
337C8—22no—
338C8—44no—
339C8—41no—
340C8—161no—
341C8—161no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22#167912yes—
351C22#168012yes—
352C22#196211yescharacteristic
353C22#196411yescharacteristic
354C22#196811yescharacteristic
355C22#197411yescharacteristic
356C22#197611yescharacteristic
357C22#197911yescharacteristic
358C22#198311yescharacteristic
359C4—21no—
360C4—21no—
361C4—22no—
362C4—22no—
363C4—21no—
364C4—21no—
365C4—48no—
366C4—41no—
367C4—161no—
368C4—161no—
369C4—161no—
370C4—161no—
371C4#150011yescharacteristic
372C4#150111yescharacteristic
373C4#150411yescharacteristic
374C4#150911yescharacteristic
375C2—22no—
376C2—22no—
377C2#971311yescharacteristic
378C2#973611yescharacteristic
379C2#975111yescharacteristic
380C2#976311yescharacteristic
381C2#978111yescharacteristic
382C2#981411yescharacteristic
383C2#984311yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#487C4semidirect
13. #1990C4semidirect
14. #1956C4semidirect
19. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#330432
C8×#952
#99092
#97112
#85952
#76682
#76652