Finite Group Databasebeta

← #100994

Subgroups — #100994

Order 512.

720 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 112 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#21902C211yesmaximal, characteristic
2#15125C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#8340C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C2×#393—21no—
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1923—24no—
11#1797C2212yes—
12#1797C2212yes—
13#1779—21no—
14C4×C22⋊C8—21no—
15C2×#332—24no—
16C2×#323—21no—
17#1446—21no—
18#1442C412yes—
19#1439C412yes—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8—21no—
24C24×C4D411yescharacteristic
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C412yes—
31C22×C22⋊C4—22no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—22no—
37C2×C4⋊C8D414yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C4×C8—22no—
43#393—42no—
44#393—42no—
45#380—44no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C412yes—
54C2×#95D411yescharacteristic
55#336—22no—
56#336—22no—
57#336—22no—
58#332—48no—
59#323—42no—
60C25C22⋊C411yescharacteristic
61C22×Q8—44no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—42no—
66C22×Q8—82no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—42no—
83C4⋊Q8—28no—
84C4⋊Q8—44no—
85C4⋊Q8—48no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—82no—
103C2×C22⋊C4D812yes—
104C2×C22⋊C4SD1612yes—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42C2×D412yes—
110C2×C42#4112yes—
111C4⋊C8—44no—
112C4⋊C8—44no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—82no—
118C22⋊C8—82no—
119C4×C8—82no—
120#95—22no—
121#95—22no—
122#95—42no—
123#95—44no—
124#95—44no—
125#95—41no—
126#95—41no—
127#95—82no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24#10212yes—
139C24C2×C22⋊C411yescharacteristic
140C2×Q8—48no—
141C2×Q8—84no—
142C2×Q8—84no—
143C2×Q8—84no—
144C2×Q8—88no—
145C2×Q8—82no—
146C2×Q8—82no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—48no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
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—81no—
188C22×C4—81no—
189C22×C4—84no—
190C22×C4#10112yes—
191C22×C4#10212yes—
192C22×C4C4×D412yes—
193C22×C4#12011yescharacteristic
194C22×C4#12112yes—
195C22×C4#12311yescharacteristic
196C22×C4C2×D811yescharacteristic
197C22×C4C2×SD1611yescharacteristic
198C22×C4#13512yes—
199C22×C4#13512yes—
200C22×C4#13712yes—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—28no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—84no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—44no—
215C22⋊C4—88no—
216C42—24no—
217C42—42no—
218C42—42no—
219C42—81no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23#38611yescharacteristic
248C23C2×#10111yescharacteristic
249C23#41311yescharacteristic
250C23C2×#10211yescharacteristic
251C23#41611yescharacteristic
252C23#41611yescharacteristic
253C23#41911yescharacteristic
254Q8—88no—
255Q8—88no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—84no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#45212yes—
302C2×C4#45512yes—
303C2×C4B(4,𝔽2)12yes—
304C2×C4#46512yes—
305C2×C4#46612yes—
306C2×C4#48512yes—
307C8—84no—
308C8—162no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#154011yescharacteristic
333C22#154411yescharacteristic
334C22#154611yescharacteristic
335C22#155411yescharacteristic
336C22#187311yescharacteristic
337C22#188511yescharacteristic
338C22#188911yescharacteristic
339C22#189311yescharacteristic
340C22#190111yescharacteristic
341C22#191011yescharacteristic
342C22#191511yescharacteristic
343C4—22no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—81no—
350C4—84no—
351C4—162no—
352C4#160614yes—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2#901611yescharacteristic
359C2#913011yescharacteristic
360C2#919411yescharacteristic
361C2#939311yescharacteristic
362C2#945311yescharacteristic
363C2#946811yescharacteristic
364C2#950611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ 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
4. #8340C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219022
#151252
#85572
#83402
#70392