Finite Group Databasebeta

← #98711

Subgroups — #98711

Order 512.

644 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 124 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#8589C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic
6#7590C211yesmaximal, characteristic
7#7588C211yesmaximal, characteristic
8C2×#557—21no—
9#2726—22no—
10#2582—21no—
11#2552—21no—
12#2546C412yes—
13#2506—22no—
14#2453—22no—
15#2446—22no—
16#2365—21no—
17Q8×C4⋊C4C412yes—
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#1842C2211yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×#333—21no—
26C2×#332—21no—
27#1296—21no—
28#1296—21no—
29#1279—22no—
30#1272—22no—
31#1271—22no—
32#1271—22no—
33#557—42no—
34#557—42no—
35#557—42no—
36#557—42no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8D411yescharacteristic
40C2×C4⋊Q8D411yescharacteristic
41C2×#125—22no—
42C2×#125—24no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4×C8—21no—
54#400—24no—
55#398—24no—
56#398—24no—
57#398—24no—
58#391—22no—
59#391—22no—
60#391—21no—
61#391—21no—
62#391D412yes—
63#389—22no—
64#389—22no—
65#389—24no—
66#389—22no—
67#389—22no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389D412yes—
73#389D412yes—
74#387—24no—
75#387—24no—
76#384—22no—
77#384C2×C411yescharacteristic
78#384D411yescharacteristic
79#384D411yescharacteristic
80C4×C4⋊C4—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2×C412yes—
86C4×C4⋊C4D412yes—
87C43C2311yescharacteristic
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336—21no—
93#336—21no—
94#333—42no—
95#332—42no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8C22⋊C412yes—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—42no—
108C4⋊Q8—44no—
109C4⋊Q8—44no—
110C4⋊Q8D812yes—
111C4⋊Q8SD1612yes—
112C4⋊Q8SD1612yes—
113C4⋊Q8Q1612yes—
114#125—48no—
115#125—48no—
116#125—44no—
117#125—44no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C42—24no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C4⋊C8—42no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4×C8—22no—
174C4×C8—41no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8#10214yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4#12011yescharacteristic
208C22×C4#12011yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12011yescharacteristic
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—48no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—84no—
253C4⋊C4#10314yes—
254C42—22no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—44no—
261C42—48no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42C2×D811yescharacteristic
267C42C2×SD1611yescharacteristic
268C42C2×SD1611yescharacteristic
269C42C2×Q1611yescharacteristic
270C42AGL(1,ℤ/8)11yescharacteristic
271C42AGL(1,ℤ/8)11yescharacteristic
272C42#13711yescharacteristic
273C42#13711yescharacteristic
274C23#37911yescharacteristic
275Q8—44no—
276Q8—44no—
277Q8—42no—
278Q8—42no—
279Q8—42no—
280Q8—42no—
281Q8—82no—
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
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—42no—
314C2×C4#41312yes—
315C2×C4C2×#10211yescharacteristic
316C2×C4C2×#10311yescharacteristic
317C2×C4#41811yescharacteristic
318C2×C4#41911yescharacteristic
319C2×C4#44711yescharacteristic
320C2×C4#44711yescharacteristic
321C2×C4#44811yescharacteristic
322C2×C4#44811yescharacteristic
323C2×C4#45011yescharacteristic
324C2×C4#45011yescharacteristic
325C2×C4#45111yescharacteristic
326C2×C4#45111yescharacteristic
327C2×C4#45411yescharacteristic
328C2×C4#45611yescharacteristic
329C8—82no—
330C8—161no—
331C8—161no—
332C8—161no—
333C22#178511yescharacteristic
334C22#178611yescharacteristic
335C22#178711yescharacteristic
336C22#179311yescharacteristic
337C22#179411yescharacteristic
338C22#179611yescharacteristic
339C22#179611yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—44no—
351C4—44no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—82no—
356C4#162711yescharacteristic
357C4#163711yescharacteristic
358C4#164911yescharacteristic
359C4#165111yescharacteristic
360C2#838711yescharacteristic
361C2#839211yescharacteristic
362C2#842911yescharacteristic
363C2#843311yescharacteristic
364C2#844711yescharacteristic
365C2#846111yescharacteristic
366C2#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
#85892
#76522
#75902
#75882