Finite Group Databasebeta

← #103727

Subgroups — #103727

Order 512.

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

normal subgroups only

This group has 523 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#29219C211yesmaximal, characteristic
2#28440C211yesmaximal, characteristic
3#28283C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic
5#9865C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#494—21no—
9C2×#475—21no—
10C2×#462—22no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14#2099—22no—
15#2066—22no—
16#2016—22no—
17#1990—21no—
18#1988—22no—
19#1981—21no—
20#1956—21no—
21#1948—21no—
22#1842C2211yescharacteristic
23#1834—21no—
24#1803—21no—
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—42no—
42#475—42no—
43#462—44no—
44#462—44no—
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—22no—
53C2×#103—24no—
54C2×#103—22no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#391—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384D411yescharacteristic
69#384D411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C43C2311yescharacteristic
75#340—22no—
76#340—24no—
77#340—24no—
78#336—21no—
79#336—21no—
80#336D412yes—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—22no—
84C2×Q16—48no—
85C2×Q16—48no—
86C2×Q16—84no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—44no—
98C4⋊Q8—41no—
99C4⋊Q8—42no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—42no—
103C4⋊Q8—41no—
104C4⋊Q8D812yes—
105C4⋊Q8Q1612yescomplemented
106C4×Q8—41no—
107C4×Q8—42no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—42no—
111C4×Q8—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
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—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139#106—44no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—42no—
143C4⋊C8—41no—
144C4⋊C8—42no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8SD1614yes—
150#103—44no—
151#103—48no—
152#103—82no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156C4×C8—42no—
157C4×C8—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4#12011yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12711yescharacteristic
188Q16—816no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—81no—
200C2×C8—81no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C42—21no—
231C42—22no—
232C42—21no—
233C42—22no—
234C42—41no—
235C42—81no—
236C42—81no—
237C42—82no—
238C42C2×D811yescharacteristic
239C42C2×SD1612yes—
240C42C2×Q1611yescharacteristic
241C42#13511yescharacteristic
242C42#13511yescharacteristic
243C42AGL(1,ℤ/8)11yescharacteristic
244C42AGL(1,ℤ/8)11yescharacteristic
245C42AGL(1,ℤ/8)12yes—
246C42#13711yescharacteristic
247C42#13711yescharacteristic
248C23#39211yescharacteristic
249Q8—44no—
250Q8—42no—
251Q8—42no—
252Q8—81no—
253Q8—82no—
254Q8—81no—
255Q8—84no—
256Q8—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4#44711yescharacteristic
282C2×C4#44711yescharacteristic
283C2×C4#44911yescharacteristic
284C2×C4#44911yescharacteristic
285C2×C4#45012yes—
286C2×C4#45111yescharacteristic
287C2×C4#45111yescharacteristic
288C2×C4#45211yescharacteristic
289C2×C4#45211yescharacteristic
290C2×C4#45512yes—
291C2×C4#45511yescharacteristic
292C2×C4#45611yescharacteristic
293C2×C4#46012yes—
294C2×C4#46611yescharacteristic
295C2×C4#46711yescharacteristic
296C2×C4#46911yescharacteristic
297C2×C4#47011yescharacteristic
298C2×C4#47312yes—
299C2×C4#49211yescharacteristic
300C2×C4#49611yescharacteristic
301C8—44no—
302C8—82no—
303C8—82no—
304C8—82no—
305C22#199911yescharacteristic
306C22#200111yescharacteristic
307C22#200411yescharacteristic
308C22#200711yescharacteristic
309C22#200911yescharacteristic
310C22#201411yescharacteristic
311C22#202011yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4—161no—
323C4—161no—
324C4#163912yes—
325C4#164111yescharacteristic
326C4#164212yes—
327C4#164411yescharacteristic
328C4#164911yescharacteristic
329C4#165211yescharacteristic
330C4#167012yes—
331C4#169012yes—
332C2#1006211yescharacteristic
333C2#1011511yescharacteristic
334C2#1018511yescharacteristic
335C2#1020611yescharacteristic
336C2#1023311yescharacteristic
337C2#1025211yescharacteristic
338C2#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
105. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#292192
#284402
#282832
#98982
#98652
#75702
#70312