Finite Group Databasebeta

← #98545

Subgroups — #98545

Order 512.

992 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2545C211yesmaximal, characteristic
2#8560C212yesmaximal, complemented
3#8342C212yesmaximal, complemented
4#7640C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C22×#124C411yescharacteristic
7C22×#124C411yescharacteristic
8C22×C4×Q8C411yescharacteristic
9#2545—24no—
10#2545—24no—
11C2×#396—22no—
12C2×#391C411yescharacteristic
13C2×#390C2211yescharacteristic
14C2×#388—22no—
15C2×C4×C4⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#333—24no—
18C2×#324—22no—
19#1280—22no—
20#1270—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2×C411yescharacteristic
23C23×D4C2×C411yescharacteristic
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—44no—
26C2×C4⋊Q8D414yes—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—22no—
30C2×#124—22no—
31C2×#124—42no—
32C2×#124—42no—
33C2×#124—44no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—42no—
37C2×C4×Q8—42no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—22no—
46C22×C42—21no—
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—24no—
51C2×C22⋊C8—24no—
52#396—216no—
53#391—22no—
54#391—22no—
55#390—22no—
56#390D414yes—
57#388—216no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—22no—
61C2×#95D412yes—
62#336—24no—
63#333—48no—
64#324—44no—
65C25—21no—
66C22×Q8—24no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C22⋊C412yes—
77C22×Q8C22⋊C412yes—
78C22×D4—21no—
79C22×D4—42no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—42no—
83C22×D4C22⋊C412yes—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C22×C8—44no—
93C4⋊Q8—28no—
94C4⋊Q8—44no—
95C4⋊Q8—48no—
96#124—48no—
97#124—48no—
98#124—48no—
99#124—44no—
100#124—44no—
101C4×Q8—48no—
102C4×Q8—48no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C22⋊C412yes—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—48no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—24no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C4⋊C8—48no—
142C22⋊C8—44no—
143C22⋊C8—44no—
144#95—28no—
145#95—24no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24—44no—
150C24—41no—
151C24—41no—
152C24—42no—
153C24—42no—
154C24C2×C22⋊C411yescharacteristic
155C2×Q8—24no—
156C2×Q8—22no—
157C2×Q8—24no—
158C2×Q8—42no—
159C2×Q8—48no—
160C2×Q8—48no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—84no—
170C2×Q8—84no—
171C2×Q8#9914yes—
172C2×D4—24no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—84no—
179C2×D4—84no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#10012yes—
214C22×C4#10012yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4#12014yes—
218C22×C4#13514yes—
219C22×C4AGL(1,ℤ/8)12yes—
220C22×C4#13712yes—
221C2×C8—84no—
222C2×C8—84no—
223C2×C8—84no—
224C4⋊C4—416no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—48no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C22⋊C4—416no—
231C22⋊C4—48no—
232C22⋊C4—48no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C42—24no—
236C42—24no—
237C42—24no—
238C42—42no—
239C42—48no—
240C42—42no—
241C42—42no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—44no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—44no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—84no—
255C23—84no—
256C23—84no—
257C23—84no—
258C23—81no—
259C23—82no—
260C23—81no—
261C23#37911yescharacteristic
262C23C2×#10011yescharacteristic
263C23C2×#10011yescharacteristic
264C23#41612yes—
265C23#41712yes—
266Q8—44no—
267Q8—44no—
268Q8—48no—
269Q8—84no—
270Q8—84no—
271D4—84no—
272D4—84no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C2×#9912yes—
314C2×C4#41312yes—
315C2×C4#44914yes—
316C2×C4#45214yes—
317C8—164no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—84no—
330C22—84no—
331C22—162no—
332C22—164no—
333C22—162no—
334C22#154311yescharacteristic
335C22#155112yes—
336C22#156011yescharacteristic
337C22#178111yescharacteristic
338C22#178311yescharacteristic
339C22#178511yescharacteristic
340C22#178812yes—
341C22#179612yes—
342C4—22no—
343C4—22no—
344C4—24no—
345C4—24no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4—82no—
350C4—82no—
351C4—82no—
352C2—21no—
353C2—22no—
354C2—21no—
355C2—162no—
356C2#834411yescharacteristic
357C2#837712yes—
358C2#839412yes—
359C2#852411yescharacteristic
360C2#854911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
2. #8560C2semidirect
3. #8342C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#85602
#83422
#76402