Finite Group Databasebeta

← #103980

Subgroups — #103980

Order 512.

443 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 90 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#29273C211yesmaximal, characteristic
3#28161C211yesmaximal, characteristic
4#9874C211yesmaximal, characteristic
5#7591C211yesmaximal, characteristic
6#7590C211yesmaximal, characteristic
7#7040C211yesmaximal, characteristic
8C2×#462—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12#2099—21no—
13#2088—22no—
14#2072—21no—
15#2058—21no—
16#2016—21no—
17#1988—21no—
18#1987—21no—
19#1987—21no—
20#1948—21no—
21#1948—21no—
22#1948—21no—
23#1919—22no—
24#1878—22no—
25#1842C2211yescharacteristic
26#1831C2211yescharacteristic
27#1831C2211yescharacteristic
28#1803—21no—
29#1744C2211yescharacteristic
30#1279—22no—
31#1272—22no—
32#1271—22no—
33C22×Q16—22no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40#462—42no—
41#462—42no—
42C2×#106—22no—
43C2×#106—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D412yes—
49C2×#103—22no—
50C2×#103—22no—
51C2×C4×C8—21no—
52#391—21no—
53#391—21no—
54#391—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#384D411yescharacteristic
62#384D411yescharacteristic
63#384D411yescharacteristic
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C43C2311yescharacteristic
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D412yes—
83#336D412yes—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—21no—
87C2×Q16—44no—
88C2×Q16—44no—
89C2×Q16—82no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100C4⋊Q8—42no—
101C4⋊Q8SD1612yes—
102C4⋊Q8Q1612yescomplemented
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—82no—
108C4×Q8—82no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
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—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42#4111yescharacteristic
137#106—44no—
138#106—82no—
139C4⋊C8—42no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148#103—44no—
149#103—82no—
150C4×C8—22no—
151C4×C8—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—81no—
171C2×Q8—81no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#12011yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12711yescharacteristic
185Q16—88no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—81no—
197C2×C8—81no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—82no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42—82no—
233C42—82no—
234C42—81no—
235C42—81no—
236C42C2×SD1611yescharacteristic
237C42C2×Q1611yescharacteristic
238C42#13511yescharacteristic
239C42#13511yescharacteristic
240C42AGL(1,ℤ/8)11yescharacteristic
241C42#13711yescharacteristic
242C42#13711yescharacteristic
243C42#13711yescharacteristic
244C23#39211yescharacteristic
245Q8—42no—
246Q8—44no—
247Q8—42no—
248Q8—82no—
249Q8—81no—
250Q8—81no—
251Q8—82no—
252Q8—82no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4#44811yescharacteristic
285C2×C4#44911yescharacteristic
286C2×C4#45011yescharacteristic
287C2×C4#45111yescharacteristic
288C2×C4#45111yescharacteristic
289C2×C4#45211yescharacteristic
290C2×C4#45211yescharacteristic
291C2×C4#45211yescharacteristic
292C2×C4#45511yescharacteristic
293C2×C4#45611yescharacteristic
294C2×C4#46511yescharacteristic
295C2×C4#46711yescharacteristic
296C2×C4#46811yescharacteristic
297C2×C4#47011yescharacteristic
298C2×C4#47112yes—
299C8—82no—
300C8—82no—
301C8—82no—
302C8—82no—
303C22#200011yescharacteristic
304C22#200111yescharacteristic
305C22#200711yescharacteristic
306C22#200811yescharacteristic
307C22#200911yescharacteristic
308C22#201011yescharacteristic
309C22#201911yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—82no—
321C4—82no—
322C4—82no—
323C4—161no—
324C4—161no—
325C4—161no—
326C4#164011yescharacteristic
327C4#164511yescharacteristic
328C4#164811yescharacteristic
329C4#165311yescharacteristic
330C2#1000011yescharacteristic
331C2#1000311yescharacteristic
332C2#1002911yescharacteristic
333C2#1018211yescharacteristic
334C2#1018911yescharacteristic
335C2#1021511yescharacteristic
336C2#1023411yescharacteristic
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
102. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#292872
#292732
#281612
#98742
#75912
#75902
#70402