Finite Group Databasebeta

← #98698

Subgroups — #98698

Order 512.

644 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 120 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#59828C211yesmaximal, characteristic
2#9874C211yesmaximal, characteristic
3#8590C211yesmaximal, characteristic
4#8590C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic
6#7590C211yesmaximal, characteristic
7#7590C211yesmaximal, characteristic
8C2×#557C412yes—
9#2726—22no—
10#2582C412yes—
11#2552—21no—
12#2546—21no—
13#2506—22no—
14#2453—22no—
15#2446—22no—
16#2365—21no—
17Q8×C4⋊C4—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#1842C2211yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×#333—21no—
26C2×#333—21no—
27#1296—21no—
28#1296—21no—
29#1279—22no—
30#1272—22no—
31#1272—22no—
32#1271—22no—
33#557—48no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8D411yescharacteristic
38C2×#125—22no—
39C2×#125—24no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4×C8—21no—
51#400—24no—
52#398—24no—
53#398—24no—
54#398—24no—
55#391—22no—
56#391—22no—
57#391—21no—
58#391D412yes—
59#391D412yes—
60#389—24no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389D412yes—
71#387—24no—
72#387—24no—
73#384—22no—
74#384C2×C411yescharacteristic
75#384D411yescharacteristic
76#384D411yescharacteristic
77C4×C4⋊C4—22no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4D412yes—
84C43C2311yescharacteristic
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#333—42no—
92#333—42no—
93C22×Q8—21no—
94C22×Q8C22⋊C412yes—
95C22×Q8C22⋊C412yes—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—24no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—48no—
108C4⋊Q8SD1612yes—
109C4⋊Q8Q1612yes—
110#125—48no—
111#125—48no—
112#125—48no—
113C4×Q8—44no—
114C4×Q8—44no—
115C4×Q8—48no—
116C4×Q8—48no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C42—24no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C4⋊C8—42no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8—41no—
163C4×C8—22no—
164C4×C8—41no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C2×Q8—24no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8#10314yes—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12011yescharacteristic
199C2×C8—42no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—84no—
241C4⋊C4#10014yes—
242C42—22no—
243C42—21no—
244C42—21no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—44no—
249C42—48no—
250C42—44no—
251C42—42no—
252C42—42no—
253C42C2×SD1611yescharacteristic
254C42C2×Q1611yescharacteristic
255C42#13511yescharacteristic
256C42#13511yescharacteristic
257C42AGL(1,ℤ/8)11yescharacteristic
258C42#13711yescharacteristic
259C42#13711yescharacteristic
260C42#13711yescharacteristic
261C23#37911yescharacteristic
262Q8—44no—
263Q8—44no—
264Q8—44no—
265Q8—44no—
266Q8—82no—
267Q8—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4C2×#10012yes—
300C2×C4C2×#10311yescharacteristic
301C2×C4#41611yescharacteristic
302C2×C4#41711yescharacteristic
303C2×C4#41911yescharacteristic
304C2×C4#44811yescharacteristic
305C2×C4#44811yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#44911yescharacteristic
308C2×C4#45111yescharacteristic
309C2×C4#45111yescharacteristic
310C2×C4#45211yescharacteristic
311C2×C4#45211yescharacteristic
312C2×C4#45411yescharacteristic
313C2×C4#45611yescharacteristic
314C8—82no—
315C8—161no—
316C8—161no—
317C8—161no—
318C22#178311yescharacteristic
319C22#178711yescharacteristic
320C22#178811yescharacteristic
321C22#179311yescharacteristic
322C22#179411yescharacteristic
323C22#179611yescharacteristic
324C22#179611yescharacteristic
325C4—22no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—44no—
336C4—44no—
337C4—82no—
338C4—82no—
339C4—82no—
340C4—82no—
341C4#162611yescharacteristic
342C4#163611yescharacteristic
343C4#164811yescharacteristic
344C4#165311yescharacteristic
345C2#839111yescharacteristic
346C2#839311yescharacteristic
347C2#843311yescharacteristic
348C2#843511yescharacteristic
349C2#844811yescharacteristic
350C2#845911yescharacteristic
351C2#849511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#598282
#98742
#85902
#76522
#75902