Finite Group Databasebeta

← #98724

Subgroups — #98724

Order 512.

556 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 116 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#59870C211yesmaximal, characteristic
2#9643C211yesmaximal, characteristic
3#8590C211yesmaximal, characteristic
4#8589C211yesmaximal, characteristic
5#7653C211yesmaximal, characteristic
6#7590C211yesmaximal, characteristic
7#7588C211yesmaximal, characteristic
8#2805—22no—
9#2581—22no—
10#2571—21no—
11#2568—21no—
12#2567C412yes—
13#2511—22no—
14#2509—22no—
15#2365—21no—
16#2355C412yes—
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×#332—21no—
27C2×Q8⋊C8—22no—
28#1296—21no—
29#1296—21no—
30#1272—22no—
31#1272—22no—
32#1271—22no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D411yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8—21no—
38C2×C4×Q8C2×C412yes—
39C2×C4×Q8D412yes—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4×C8—21no—
46#400—24no—
47#400—24no—
48#400—24no—
49#395—24no—
50#391—22no—
51#391—22no—
52#391—22no—
53#391—21no—
54#391D412yes—
55#391D412yes—
56#389—22no—
57#389—24no—
58#389—22no—
59#389—22no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389D412yes—
66#387—24no—
67#387—24no—
68#387—24no—
69#384—22no—
70#384C2×C411yescharacteristic
71#384D411yescharacteristic
72#384D411yescharacteristic
73#382—22no—
74#382—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2×C412yes—
83C43C2311yescharacteristic
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#333—42no—
91#332—42no—
92Q8⋊C8—44no—
93C22×Q8—21no—
94C22×Q8C22⋊C412yes—
95C22×Q8C22⋊C412yes—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—42no—
107C4⋊Q8SD1612yes—
108C4⋊Q8Q1612yes—
109C4×Q8—44no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—82no—
117C4×Q8—82no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22⋊C412yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—24no—
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C4⋊C8—42no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4×C8—22no—
159C4×C8—41no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12011yescharacteristic
198C2×C8—42no—
199C2×C8—81no—
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—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4#10314yes—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—44no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—84no—
249C42C2×SD1611yescharacteristic
250C42C2×Q1611yescharacteristic
251C42#13511yescharacteristic
252C42#13511yescharacteristic
253C42AGL(1,ℤ/8)11yescharacteristic
254C42#13711yescharacteristic
255C42#13711yescharacteristic
256C42#13711yescharacteristic
257C23#37911yescharacteristic
258Q8—44no—
259Q8—42no—
260Q8—42no—
261Q8—42no—
262Q8—42no—
263Q8—82no—
264Q8—82no—
265Q8—82no—
266C2×C4—21no—
267C2×C4—21no—
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—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
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#41312yes—
298C2×C4C2×#10311yescharacteristic
299C2×C4#41611yescharacteristic
300C2×C4#41711yescharacteristic
301C2×C4#41911yescharacteristic
302C2×C4#44811yescharacteristic
303C2×C4#44811yescharacteristic
304C2×C4#44911yescharacteristic
305C2×C4#44911yescharacteristic
306C2×C4#45111yescharacteristic
307C2×C4#45111yescharacteristic
308C2×C4#45211yescharacteristic
309C2×C4#45211yescharacteristic
310C2×C4#45311yescharacteristic
311C2×C4#45511yescharacteristic
312C8—82no—
313C8—161no—
314C8—161no—
315C8—161no—
316C22#178511yescharacteristic
317C22#178711yescharacteristic
318C22#178811yescharacteristic
319C22#179211yescharacteristic
320C22#179411yescharacteristic
321C22#179611yescharacteristic
322C22#179611yescharacteristic
323C4—22no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—41no—
334C4—82no—
335C4—82no—
336C4—82no—
337C4—82no—
338C4—82no—
339C4#162211yescharacteristic
340C4#163211yescharacteristic
341C4#164311yescharacteristic
342C4#164411yescharacteristic
343C2#838911yescharacteristic
344C2#839211yescharacteristic
345C2#843111yescharacteristic
346C2#843311yescharacteristic
347C2#845011yescharacteristic
348C2#845811yescharacteristic
349C2#849211yescharacteristic
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
#598702
#96432
#85902
#85892
#76532
#75902
#75882