Finite Group Databasebeta

← #98496

Subgroups — #98496

Order 512.

784 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2552C211yesmaximal, characteristic
2#8335C212yesmaximal
3#8153C211yesmaximal, characteristic
4#8153C211yesmaximal, characteristic
5#7627C212yesmaximal
6C22×#125C411yescharacteristic
7C22×#125C2211yescharacteristic
8#2552—24no—
9#2552—24no—
10C2×#398—21no—
11C2×#398—21no—
12C2×#389—21no—
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#384—21no—
16C2×#384C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1749C2212yes—
20#1749C2212yes—
21C2×#330—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C2×#125—22no—
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—44no—
31C2×#125—44no—
32C2×#125D414yes—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C42—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×#97—22no—
54C2×#97—22no—
55#398—28no—
56#398—28no—
57#389—28no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389D414yes—
62#384—28no—
63#384—22no—
64#384—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#330—48no—
77#324—24no—
78#324—24no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—42no—
91C22×C8—42no—
92#125—28no—
93#125—44no—
94#125—48no—
95#125—48no—
96#125—44no—
97#125—44no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—24no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150#97—44no—
151#97—44no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—22no—
156#95—22no—
157C24C2×C22⋊C411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4#10112yes—
209C22×C4#10112yes—
210C22×C4#10412yes—
211C22×C4#10412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4#12012yes—
215C22×C4#12012yes—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C4⋊C4—42no—
232C4⋊C4—48no—
233C4⋊C4—42no—
234C4⋊C4—48no—
235C4⋊C4—48no—
236C4⋊C4—42no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C4⋊C4#9914yes—
241C4⋊C4#10114yes—
242C42—24no—
243C42—24no—
244C42—24no—
245C42—42no—
246C42—48no—
247C42—42no—
248C42—42no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23#37911yescharacteristic
254C23C2×#10111yescharacteristic
255C23C2×#10111yescharacteristic
256C23C2×#10411yescharacteristic
257C23C2×#10411yescharacteristic
258C23#42111yescharacteristic
259C23#42111yescharacteristic
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#33014yescomplemented
318C2×C4C2×#9912yes—
319C2×C4C2×#10112yes—
320C8—84no—
321C8—84no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22C2×#33011yescharacteristic
335C22#152811yescharacteristic
336C22#152811yescharacteristic
337C22#153411yescharacteristic
338C22#178111yescharacteristic
339C22#178411yescharacteristic
340C22#178411yescharacteristic
341C22#178911yescharacteristic
342C22#178911yescharacteristic
343C22#179511yescharacteristic
344C22#179611yescharacteristic
345C4—22no—
346C4—22no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—44no—
352C4—44no—
353C4—81no—
354C4—82no—
355C4—82no—
356C4—81no—
357C2—21no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2#833511yescharacteristic
362C2#836111yescharacteristic
363C2#836411yescharacteristic
364C2#837211yescharacteristic
365C2#837411yescharacteristic
366C2#852511yescharacteristic
367C2#855111yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊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
317. C2×C4#330semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#83352
#81532
#76272