Finite Group Databasebeta

← #99650

Subgroups — #99650

Order 512.

685 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×#2096C211yesmaximal, characteristic
2#14467C211yesmaximal, characteristic
3#11357C212yesmaximal
4C2×#1272C211yesmaximal, characteristic
5#7025C212yesmaximal
6C22×#106—21no—
7C22×#103—21no—
8C2×#391—21no—
9C2×#391C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2096—22no—
12#2096C414yescomplemented
13C2×#340—22no—
14C2×#340—22no—
15C2×#336C2212yes—
16C2×#336C2211yescharacteristic
17#1444—22no—
18#1441—21no—
19#1436—22no—
20C4⋊#95C2212yes—
21#1272—22no—
22C23×Q8D411yescharacteristic
23C23×C8—21no—
24C23×C8—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4D412yes—
32C22×C42—21no—
33C22×C42C2311yescharacteristic
34C2×#106—24no—
35C2×#106—42no—
36C2×#106—42no—
37C2×#103—24no—
38C2×#103—42no—
39C2×#103—42no—
40#391—22no—
41#391—42no—
42#391—42no—
43#391C2×C412yes—
44#391D412yes—
45#384—21no—
46#384—22no—
47#384C2×C412yes—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—22no—
54#340—44no—
55#340—44no—
56#336—21no—
57#336—22no—
58#336—22no—
59#336C2×C412yes—
60C22×Q8—21no—
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8C22⋊C412yes—
68C22×Q8SD1612yes—
69C22×Q8Q1612yes—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—41no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—24no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—81no—
115C2×C4⋊C4—81no—
116C2×C4⋊C4—81no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4SD1614yes—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—42no—
123C2×C42—81no—
124C2×C42—81no—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42#4112yes—
128#106—48no—
129#103—48no—
130#95—24no—
131#95—24no—
132#95—44no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—81no—
138#95—82no—
139C24#12011yescharacteristic
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—81no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8#10314yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C4×D412yes—
197C22×C4#12112yes—
198C22×C4#12311yescharacteristic
199C22×C4C2×SD1611yescharacteristic
200C22×C4C2×SD1612yes—
201C22×C4C2×Q1611yescharacteristic
202C22×C4#13512yes—
203C22×C4AGL(1,ℤ/8)11yescharacteristic
204C22×C4AGL(1,ℤ/8)12yes—
205C22×C4#13711yescharacteristic
206C22×C4#13712yes—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—82no—
229C4⋊C4—81no—
230C4⋊C4—82no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C42—41no—
238C42—41no—
239C42—42no—
240C42—84no—
241C23#38611yescharacteristic
242C23#44812yes—
243C23#44912yes—
244C23#45011yescharacteristic
245C23#45111yescharacteristic
246C23#45311yescharacteristic
247C23#45311yescharacteristic
248C23#45512yes—
249C23#45612yes—
250C23#45811yescharacteristic
251C23#45811yescharacteristic
252Q8—44no—
253Q8—82no—
254Q8—82no—
255Q8—82no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—162no—
309C2×C4—161no—
310C2×C4—161no—
311C2×C4C2×#10311yescharacteristic
312C2×C4#41711yescharacteristic
313C2×C4C4×SD1612yes—
314C2×C4#44012yes—
315C2×C4#46012yes—
316C2×C4#46412yes—
317C2×C4#48311yescharacteristic
318C2×C4#48411yescharacteristic
319C8—44no—
320C8—84no—
321C22#160812yes—
322C22#160912yes—
323C22#161112yes—
324C22#161212yes—
325C22#161311yescharacteristic
326C22#161311yescharacteristic
327C22#161411yescharacteristic
328C22#161411yescharacteristic
329C22#162212yes—
330C22#162311yescharacteristic
331C22#162411yescharacteristic
332C22#164212yes—
333C22#164512yes—
334C22#164812yes—
335C22#165112yes—
336C22#165911yescharacteristic
337C22#166011yescharacteristic
338C22#166112yes—
339C22#187211yescharacteristic
340C22#188012yes—
341C22#188212yes—
342C22#188711yescharacteristic
343C22#189011yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—44no—
353C4—82no—
354C4—84no—
355C4—162no—
356C4—162no—
357C2#780112yes—
358C2#780211yescharacteristic
359C2#780912yes—
360C2#781011yescharacteristic
361C2#781211yescharacteristic
362C2#781711yescharacteristic
363C2#898312yes—
364C2#900311yescharacteristic
365C2#902111yescharacteristic
366C2#904412yes—
367C2#906311yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2096C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
#144672
#113572
C2×#12722
#70252