Finite Group Databasebeta

← #114114

Subgroups — #114114

Order 512.

1160 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 194 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×#2503C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3C2×#1520C212yesmaximal
4#10846C211yesmaximal, characteristic
5#7907C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2503—24no—
10#2503—24no—
11C2×#400—22no—
12C2×#391C412yescomplemented
13C2×#382C412yescomplemented
14C2×#381—22no—
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1823—24no—
18#1823—24no—
19C2×#342—22no—
20C2×#340—24no—
21#1520—24no—
22#1520—24no—
23C2×#328—24no—
24#1280—22no—
25C23×Q8C2×C411yescharacteristic
26C24×C4—21no—
27C24×C4C2311yescharacteristic
28C23×C8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8D414yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D414yes—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4D412yes—
49C22×C42C2×C412yes—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8D414yes—
56#400—216no—
57#391—24no—
58#391—24no—
59#382—24no—
60#382—24no—
61#381—216no—
62#380—22no—
63#380—22no—
64#380—22no—
65#380D414yes—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—22no—
71C2×#95D412yes—
72#342—44no—
73#340—216no—
74#328—216no—
75C25C2×D411yescharacteristic
76C22×Q8—21no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8—44no—
80C22×Q8C22⋊C412yes—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—24no—
85C23×C4—44no—
86C23×C4—42no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—44no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4C22⋊C414yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4#4112yes—
101C23×C4#4112yes—
102C22×C8—24no—
103C22×C8—48no—
104C22×C8—42no—
105C22×C8—42no—
106C22⋊Q8—24no—
107C22⋊Q8—24no—
108C22⋊Q8—24no—
109C22⋊Q8—24no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112C22⋊Q8—42no—
113C22⋊Q8—42no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—44no—
156C22⋊C8—24no—
157C22⋊C8—24no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160#95—216no—
161#95—216no—
162#95—24no—
163#95—24no—
164#95—42no—
165C24—42no—
166C24—42no—
167C24C2×C22⋊C411yescharacteristic
168C24#12012yes—
169C24#12112yes—
170C24#12411yescharacteristic
171C24#12711yescharacteristic
172C24#13514yes—
173C24AGL(1,ℤ/8)12yes—
174C24#13712yes—
175C2×Q8—24no—
176C2×Q8—44no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—84no—
180C2×Q8—84no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C4×D412yes—
225C22×C4C4×D412yes—
226C22×C4#12012yes—
227C22×C4#12012yes—
228C22×C4#12012yes—
229C22×C4#12112yes—
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4#12112yes—
233C22×C4#12112yes—
234C22×C4#12312yes—
235C22×C4#12312yes—
236C22×C4#12411yescharacteristic
237C22×C4#12711yescharacteristic
238C2×C8—48no—
239C2×C8—44no—
240C2×C8—84no—
241C2×C8—84no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—84no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C42—44no—
260C42—44no—
261C23—22no—
262C23—24no—
263C23—22no—
264C23—24no—
265C23—42no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—48no—
272C23—44no—
273C23—44no—
274C23#37911yescharacteristic
275C23#39011yescharacteristic
276C23#39011yescharacteristic
277C23#39212yes—
278C23#39412yes—
279C23#41612yes—
280C23#41712yes—
281C23#44914yes—
282C23#45214yes—
283C23#46314yes—
284C23#46414yes—
285C23#48812yes—
286C23#48912yes—
287C23#49512yes—
288C23#49611yescharacteristic
289C23#49711yescharacteristic
290Q8—84no—
291Q8—84no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—168no—
332C2×C4#38612yes—
333C2×C4#38612yes—
334C2×C4#39212yes—
335C2×C4#39412yes—
336C2×C4#41014yes—
337C8—84no—
338C22—24no—
339C22—24no—
340C22—28no—
341C22—28no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—44no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22#143811yescharacteristic
358C22#178212yes—
359C22#178812yes—
360C22#179612yes—
361C22#195811yescharacteristic
362C22#195911yescharacteristic
363C22#198312yes—
364C22#201212yes—
365C22#201512yes—
366C22#204214yes—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—84no—
371C4—84no—
372C4—82no—
373C4—82no—
374C4—164no—
375C4#189914yes—
376C2—22no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—22no—
381C2—24no—
382C2#838214yes—
383C2#1096611yescharacteristic
384C2#1100711yescharacteristic
385C2#1100911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#391C4semidirect
13. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#18232
C2×#15202
#108462
#79072