Finite Group Databasebeta

← #103723

Subgroups — #103723

Order 512.

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

normal subgroups only

This group has 541 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#29287C211yesmaximal, characteristic
2#28512C211yesmaximal, characteristic
3#28440C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic
5#9864C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#494—21no—
9C2×#475—22no—
10C2×#462—22no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14#2096—22no—
15#2088—22no—
16#2016—22no—
17#1988—21no—
18#1987—21no—
19#1956—21no—
20#1956—21no—
21#1948—21no—
22#1948—21no—
23#1844—21no—
24#1842C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1296—22no—
29#1279—22no—
30#1271—22no—
31C22×Q16—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D412yes—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39#494—42no—
40#494—42no—
41#475—48no—
42#462—44no—
43#462—44no—
44C2×#106—24no—
45C2×#106—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×#103—24no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391—21no—
58#391—21no—
59#389—21no—
60#389—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—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336D412yes—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—22no—
83C2×Q16—48no—
84C2×Q16—48no—
85C2×Q16—84no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—44no—
97C4⋊Q8—42no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—42no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8SD1612yes—
104C4⋊Q8Q1612yescomplemented
105C4×Q8—44no—
106C4×Q8—42no—
107C4×Q8—42no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42#4111yescharacteristic
135#106—48no—
136#106—82no—
137C4⋊C8—24no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—42no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8SD1614yes—
147#103—48no—
148#103—48no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—42no—
153C4×C8—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4#12011yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12711yescharacteristic
182Q16—816no—
183C2×C8—24no—
184C2×C8—24no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—81no—
194C2×C8—81no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—82no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—22no—
227C42—41no—
228C42—82no—
229C42—82no—
230C42C2×SD1611yescharacteristic
231C42C2×SD1612yes—
232C42C2×Q1611yescharacteristic
233C42#13511yescharacteristic
234C42#13511yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C42AGL(1,ℤ/8)12yes—
237C42#13711yescharacteristic
238C42#13711yescharacteristic
239C42#13711yescharacteristic
240C23#39211yescharacteristic
241Q8—44no—
242Q8—44no—
243Q8—82no—
244Q8—82no—
245Q8—84no—
246Q8—84no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4#44811yescharacteristic
272C2×C4#44911yescharacteristic
273C2×C4#45011yescharacteristic
274C2×C4#45012yes—
275C2×C4#45111yescharacteristic
276C2×C4#45111yescharacteristic
277C2×C4#45211yescharacteristic
278C2×C4#45211yescharacteristic
279C2×C4#45211yescharacteristic
280C2×C4#45312yes—
281C2×C4#45511yescharacteristic
282C2×C4#45611yescharacteristic
283C2×C4#46012yes—
284C2×C4#46511yescharacteristic
285C2×C4#46711yescharacteristic
286C2×C4#46811yescharacteristic
287C2×C4#47011yescharacteristic
288C2×C4#47312yes—
289C2×C4#49211yescharacteristic
290C2×C4#49611yescharacteristic
291C8—44no—
292C8—44no—
293C8—82no—
294C8—82no—
295C22#200011yescharacteristic
296C22#200111yescharacteristic
297C22#200211yescharacteristic
298C22#200711yescharacteristic
299C22#200911yescharacteristic
300C22#201411yescharacteristic
301C22#202011yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4—162no—
312C4—161no—
313C4—161no—
314C4#162912yes—
315C4#163112yes—
316C4#164011yescharacteristic
317C4#164511yescharacteristic
318C4#164811yescharacteristic
319C4#165311yescharacteristic
320C4#167012yes—
321C4#169012yes—
322C2#1005411yescharacteristic
323C2#1010711yescharacteristic
324C2#1018711yescharacteristic
325C2#1020611yescharacteristic
326C2#1023411yescharacteristic
327C2#1024811yescharacteristic
328C2#1027611yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
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
104. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#292872
#285122
#284402
#98982
#98642
#75702
#70312