Finite Group Databasebeta

← #98498

Subgroups — #98498

Order 512.

816 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2550C211yesmaximal, characteristic
2#8335C212yesmaximal
3#8153C211yesmaximal, characteristic
4#8153C211yesmaximal, characteristic
5#7627C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×#125C2211yescharacteristic
8#2550—24no—
9#2550—24no—
10C2×#398—21no—
11C2×#395—21no—
12C2×#391—21no—
13C2×#389—21no—
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#382C411yescharacteristic
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—
27C23×Q8—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—44no—
31C2×#125—22no—
32C2×#125—44no—
33C2×#125D414yes—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×#97—22no—
52C2×#97—22no—
53#398—28no—
54#395—28no—
55#391—28no—
56#389—28no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#382—22no—
62#382—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C2×#95—21no—
68C2×#95—21no—
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—
79C22×Q8—24no—
80C22×Q8—44no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8—82no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—42no—
96C22×C8—42no—
97C4⋊Q8—48no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100#125—28no—
101#125—44no—
102#125—48no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—24no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—82no—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148#97—44no—
149#97—44no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—22no—
156#95—22no—
157C24C2×C22⋊C411yescharacteristic
158C2×Q8—48no—
159C2×Q8—48no—
160C2×Q8—84no—
161C2×Q8—88no—
162C2×Q8—88no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4#10112yes—
214C22×C4#10112yes—
215C22×C4#10412yes—
216C22×C4#10412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4#12012yes—
220C22×C4#12012yes—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—24no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—48no—
236C4⋊C4—42no—
237C4⋊C4—48no—
238C4⋊C4—42no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4#9914yes—
243C4⋊C4#10114yes—
244C42—24no—
245C42—24no—
246C42—24no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—84no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#37911yescharacteristic
256C23C2×#10111yescharacteristic
257C23C2×#10111yescharacteristic
258C23C2×#10411yescharacteristic
259C23C2×#10411yescharacteristic
260C23#42111yescharacteristic
261C23#42111yescharacteristic
262Q8—816no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4C2×#9912yes—
322C2×C4C2×#10112yes—
323C8—84no—
324C8—84no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#152611yescharacteristic
338C22#153111yescharacteristic
339C22#153111yescharacteristic
340C22#153711yescharacteristic
341C22#178111yescharacteristic
342C22#178411yescharacteristic
343C22#178411yescharacteristic
344C22#178911yescharacteristic
345C22#178911yescharacteristic
346C22#179511yescharacteristic
347C22#179611yescharacteristic
348C4—22no—
349C4—22no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—44no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—44no—
358C4—82no—
359C4—82no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2#833811yescharacteristic
365C2#835911yescharacteristic
366C2#836611yescharacteristic
367C2#837211yescharacteristic
368C2#837411yescharacteristic
369C2#852511yescharacteristic
370C2#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

Maximal subgroups

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

subgroupindex
C2×#25502
#83352
#81532
#76272