Finite Group Databasebeta

← #103741

Subgroups — #103741

Order 512.

529 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 110 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#29218C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28252C211yesmaximal, characteristic
4#9865C211yesmaximal, characteristic, complemented
5#9691C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic, complemented
8C2×#460—21no—
9C2×#461—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2100—22no—
14#2055—22no—
15#2023—22no—
16#1990—21no—
17#1987—22no—
18#1957—21no—
19#1842C2211yescharacteristic
20#1834—21no—
21#1834—21no—
22#1803—21no—
23#1802—22no—
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#333—22no—
29#1279—22no—
30#1270—22no—
31C22×SD16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41#461—42no—
42#461—42no—
43#460—42no—
44#460—42no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D412yes—
50C2×C4⋊C8D412yes—
51C2×#103—22no—
52C2×#102—22no—
53C2×#102—24no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391—21no—
58#390—21no—
59#390—21no—
60#390—21no—
61#389—21no—
62#389—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384D411yescharacteristic
68#384D411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—24no—
74#340—24no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#333—44no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—21no—
85C23×C4—41no—
86C2×SD16—44no—
87C2×SD16—44no—
88C2×SD16—82no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—44no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—82no—
100C4⋊Q8SD1612yes—
101#127—21no—
102#127—21no—
103#127—21no—
104#127SD1612yes—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—42no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—82no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—41no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—42no—
146C4⋊C8—41no—
147C4⋊C8—42no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152#103—44no—
153#102—44no—
154#102—84no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—42no—
158C4×C8—41no—
159C4×C8—41no—
160#95—41no—
161#95—41no—
162C24—41no—
163C2×Q8—22no—
164C2×Q8—44no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—82no—
171C2×Q8—82no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—81no—
185C2×D4—81no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—82no—
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12711yescharacteristic
201SD16—88no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—81no—
246C42—81no—
247C42C2×SD1611yescharacteristic
248C42C2×SD1611yescharacteristic
249C42#13511yescharacteristic
250C42#13511yescharacteristic
251C42#13512yes—
252C42AGL(1,ℤ/8)11yescharacteristic
253C42AGL(1,ℤ/8)11yescharacteristic
254C42#13711yescharacteristic
255C42#13711yescharacteristic
256C42#13712yes—
257C23—42no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23#39211yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—81no—
268Q8—81no—
269Q8—164no—
270D4—42no—
271D4—42no—
272D4—44no—
273D4—82no—
274D4—82no—
275D4—81no—
276D4—81no—
277D4—82no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—162no—
300C2×C4—162no—
301C2×C4#44811yescharacteristic
302C2×C4#44811yescharacteristic
303C2×C4#44911yescharacteristic
304C2×C4#44911yescharacteristic
305C2×C4#45011yescharacteristic
306C2×C4#45011yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45211yescharacteristic
309C2×C4#45212yes—
310C2×C4#45311yescharacteristic
311C2×C4#45411yescharacteristic
312C2×C4#45512yes—
313C2×C4#46412yes—
314C2×C4#46511yescharacteristic
315C2×C4#46511yescharacteristic
316C2×C4#46811yescharacteristic
317C2×C4#46811yescharacteristic
318C2×C4#47212yes—
319C2×C4#49511yescharacteristic
320C2×C4#49711yescharacteristic
321C8—82no—
322C8—82no—
323C8—82no—
324C8—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22#200011yescharacteristic
333C22#200011yescharacteristic
334C22#200411yescharacteristic
335C22#200611yescharacteristic
336C22#200911yescharacteristic
337C22#201511yescharacteristic
338C22#201811yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—82no—
347C4—161no—
348C4—162no—
349C4—161no—
350C4#162911yescharacteristic
351C4#163411yescharacteristic
352C4#163511yescharacteristic
353C4#163511yescharacteristic
354C4#164312yes—
355C4#164612yes—
356C4#167812yes—
357C4#169812yes—
358C2—162no—
359C2#1006411yescharacteristic
360C2#1011211yescharacteristic
361C2#1018811yescharacteristic
362C2#1020211yescharacteristic
363C2#1023511yescharacteristic
364C2#1024411yescharacteristic
365C2#1026911yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29218C2semidirect
4. #9865C2semidirect
5. #9691C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#292182
#291392
#282522
#98652
#96912
#75692
#70312