Finite Group Databasebeta

← #103720

Subgroups — #103720

Order 512.

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

normal subgroups only

This group has 559 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#28748C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic
3#28280C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7571C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#487—21no—
9C2×#476—21no—
10C2×#474—22no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×#127C2211yescharacteristic
14#2093—22no—
15#2065—22no—
16#2043—22no—
17#1982—21no—
18#1981—22no—
19#1957—21no—
20#1956—21no—
21#1948—21no—
22#1845—21no—
23#1842C2211yescharacteristic
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29#1296—22no—
30#1272—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D411yescharacteristic
34C2×C4⋊Q8D412yes—
35C2×#127D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#487—42no—
41#487—42no—
42#476—42no—
43#476—42no—
44#474—48no—
45C2×#107—24no—
46C2×#106—22no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D412yes—
52C2×C4⋊C8D412yes—
53C2×#103—24no—
54C2×#103—24no—
55C2×#102—22no—
56C2×#102—24no—
57C2×#102—22no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391—21no—
62#390—21no—
63#390—21no—
64#390—21no—
65#389—21no—
66#389—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384—21no—
71#384D411yescharacteristic
72#384D411yescharacteristic
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C43C2311yescharacteristic
76#340—22no—
77#336—21no—
78#336—21no—
79#336D412yes—
80D4⋊C8—44no—
81C22×Q8—22no—
82C22×Q8—21no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—21no—
86C23×C4—41no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—44no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—42no—
98C4⋊Q8—41no—
99C4⋊Q8—42no—
100C4⋊Q8—41no—
101C4⋊Q8D812yes—
102#127—21no—
103#127—21no—
104#127—21no—
105#127Q1612yes—
106C4×Q8—82no—
107C4×D4—44no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—42no—
111C4×D4—41no—
112C4×D4—42no—
113C4×D4—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C42—41no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140#107—48no—
141#106—44no—
142C4⋊C8—24no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—42no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152#103—48no—
153#103—84no—
154#102—44no—
155#102—48no—
156#102—82no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—42no—
161C4×C8—41no—
162#95—41no—
163#95—41no—
164C24—41no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4#12011yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12711yescharacteristic
202C2×C8—24no—
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—44no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—82no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C22⋊C4—82no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—22no—
242C42—21no—
243C42—21no—
244C42—22no—
245C42—41no—
246C42—82no—
247C42C2×D811yescharacteristic
248C42C2×Q1611yescharacteristic
249C42#13511yescharacteristic
250C42#13512yes—
251C42#13511yescharacteristic
252C42AGL(1,ℤ/8)11yescharacteristic
253C42AGL(1,ℤ/8)11yescharacteristic
254C42AGL(1,ℤ/8)12yes—
255C42#13711yescharacteristic
256C42#13711yescharacteristic
257C23—42no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23#39211yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—84no—
268Q8—84no—
269D4—44no—
270D4—42no—
271D4—42no—
272D4—44no—
273D4—82no—
274D4—81no—
275D4—81no—
276D4—82no—
277C2×C4—21no—
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—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—162no—
299C2×C4—162no—
300C2×C4#44711yescharacteristic
301C2×C4#44711yescharacteristic
302C2×C4#44911yescharacteristic
303C2×C4#44912yes—
304C2×C4#44911yescharacteristic
305C2×C4#45111yescharacteristic
306C2×C4#45111yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45211yescharacteristic
309C2×C4#45312yes—
310C2×C4#45511yescharacteristic
311C2×C4#45611yescharacteristic
312C2×C4#46312yes—
313C2×C4#46611yescharacteristic
314C2×C4#46711yescharacteristic
315C2×C4#46911yescharacteristic
316C2×C4#47011yescharacteristic
317C2×C4#47112yes—
318C2×C4#49511yescharacteristic
319C2×C4#49611yescharacteristic
320C8—44no—
321C8—82no—
322C8—82no—
323C8—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22#199911yescharacteristic
332C22#200111yescharacteristic
333C22#200211yescharacteristic
334C22#200711yescharacteristic
335C22#200911yescharacteristic
336C22#201211yescharacteristic
337C22#201911yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—82no—
346C4—161no—
347C4—162no—
348C4—161no—
349C4#162012yes—
350C4#162112yes—
351C4#164111yescharacteristic
352C4#164411yescharacteristic
353C4#164911yescharacteristic
354C4#165211yescharacteristic
355C4#168312yes—
356C4#170312yes—
357C2—162no—
358C2#1004711yescharacteristic
359C2#1010611yescharacteristic
360C2#1018411yescharacteristic
361C2#1020511yescharacteristic
362C2#1023311yescharacteristic
363C2#1024811yescharacteristic
364C2#1027511yescharacteristic
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. #28748C2semidirect
4. #9898C2semidirect
6. #7571C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#287482
#285082
#282802
#98982
#96742
#75712
#70312