Finite Group Databasebeta

← #103603

Subgroups — #103603

Order 512.

674 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 100 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#29966C211yesmaximal, characteristic
2#25722C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#9650C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7665C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#494—21no—
10C2×#451—22no—
11C22×#107—21no—
12#2857C2211yescharacteristic
13C2×#389C2211yescharacteristic
14#2099—21no—
15#2086—22no—
16#2057—22no—
17#2057—21no—
18#2057—22no—
19#2016—22no—
20#1956—21no—
21#1852—21no—
22#1823—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#332—22no—
29C2×#324—22no—
30#1271—22no—
31C23×Q8—21no—
32C22×Q16—24no—
33C23×C8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—44no—
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8D412yes—
40C2×C22⋊Q8—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C42C2311yescharacteristic
47#494—44no—
48#451—44no—
49#451—44no—
50C2×#107—24no—
51C2×#107—22no—
52C2×#107—44no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D412yes—
55C2×#103—24no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8D412yes—
59C2×C4×C8—21no—
60#391—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#384D411yescharacteristic
68#380—21no—
69#380—21no—
70C2×#95—21no—
71#340—24no—
72#340—22no—
73#340—22no—
74#340—22no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#332—44no—
81#324—44no—
82C22×Q8—22no—
83C22×Q8—21no—
84C22×Q8—44no—
85C22×Q8—41no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—82no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C2×Q16—48no—
96C2×Q16—48no—
97C2×Q16—84no—
98C22×C8—24no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4Q1612yescomplemented
141C2×C22⋊C4—41no—
142C2×C42—42no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147#107—48no—
148#107—82no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151#103—48no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—42no—
156C22⋊C8D814yes—
157C4×C8—42no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162C24#12111yescharacteristic
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8—44no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—88no—
173C2×Q8—88no—
174C2×Q8—81no—
175C2×Q8—81no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4#12011yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4C2×D812yes—
205C22×C4C2×Q1611yescharacteristic
206C22×C4#13511yescharacteristic
207C22×C4AGL(1,ℤ/8)12yes—
208C22×C4#13711yescharacteristic
209C22×C4#13711yescharacteristic
210Q16—816no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C22⋊C4—82no—
248C42—22no—
249C42—22no—
250C42—41no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#39211yescharacteristic
257C23#45912yes—
258C23#46711yescharacteristic
259C23#47011yescharacteristic
260C23#47312yes—
261Q8—48no—
262Q8—84no—
263Q8—84no—
264Q8—84no—
265Q8—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—88no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—162no—
303C2×C4#44712yes—
304C2×C4#45111yescharacteristic
305C2×C4#45111yescharacteristic
306C2×C4#45211yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45612yes—
309C2×C4B(4,𝔽2)12yes—
310C2×C4#47312yes—
311C2×C4#49311yescharacteristic
312C2×C4#49611yescharacteristic
313C8—44no—
314C8—44no—
315C8—82no—
316C8—161no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22#166912yes—
326C22#168912yes—
327C22#200111yescharacteristic
328C22#200511yescharacteristic
329C22#200811yescharacteristic
330C22#201111yescharacteristic
331C22#202011yescharacteristic
332C22#202011yescharacteristic
333C22#202111yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—82no—
342C4—84no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4#160312yes—
347C4#160512yes—
348C2—22no—
349C2—22no—
350C2#992511yescharacteristic
351C2#994611yescharacteristic
352C2#999111yescharacteristic
353C2#1013911yescharacteristic
354C2#1017411yescharacteristic
355C2#1018311yescharacteristic
356C2#1027411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
140. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#299662
#257222
C2×#20572
#96502
#83402
#76652
#70292