Finite Group Databasebeta

← #103598

Subgroups — #103598

Order 512.

630 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 96 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#29757C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4#9879C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10C2×#484—21no—
11C22×#106—21no—
12#2857C2211yescharacteristic
13C2×#389C2211yescharacteristic
14#2083—21no—
15#2068—22no—
16#2058—22no—
17#2058—22no—
18#2043—22no—
19#2043—21no—
20#1956—21no—
21#1872—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1787—22no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#332—22no—
29#1280—22no—
30#1271—22no—
31C23×Q8—21no—
32C23×C8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8D411yescharacteristic
39C2×C22⋊Q8—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#487—44no—
47#484—42no—
48#484—42no—
49C2×#107—22no—
50C2×#106—24no—
51C2×#106—44no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D412yes—
54C2×#103—22no—
55C2×#103—24no—
56C2×#103—22no—
57C2×#103—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8D412yes—
61C2×C4×C8—21no—
62#391—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#384D411yescharacteristic
70#380—21no—
71#380—21no—
72C2×#95—21no—
73#340—24no—
74#340—22no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#332—44no—
81C22×Q8—22no—
82C22×Q8—21no—
83C22×Q8—44no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—82no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—42no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112C22⋊Q8—41no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
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—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4SD1612yes—
138C2×C22⋊C4—41no—
139C2×C42—42no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144#107—44no—
145#106—48no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#103—44no—
149#103—48no—
150#103—84no—
151#103—82no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—42no—
157C4×C8—42no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162C24#12111yescharacteristic
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8—44no—
167C2×Q8—41no—
168C2×Q8—44no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—82no—
173C2×Q8—88no—
174C2×Q8—88no—
175C2×Q8—81no—
176C2×Q8—81no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—44no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4#12011yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12711yescharacteristic
205C22×C4C2×SD1611yescharacteristic
206C22×C4#13511yescharacteristic
207C22×C4#13512yes—
208C22×C4AGL(1,ℤ/8)11yescharacteristic
209C22×C4#13711yescharacteristic
210C22×C4#13712yes—
211C2×C8—24no—
212C2×C8—42no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—44no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—81no—
243C4⋊C4—82no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—82no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—22no—
250C42—22no—
251C42—41no—
252C42—41no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#39211yescharacteristic
258C23#46412yes—
259C23#46511yescharacteristic
260C23#46811yescharacteristic
261C23#47312yes—
262Q8—48no—
263Q8—84no—
264Q8—84no—
265Q8—84no—
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—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—88no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—162no—
305C2×C4#44811yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#45011yescharacteristic
308C2×C4#45212yes—
309C2×C4#45211yescharacteristic
310C2×C4#45612yes—
311C2×C4B(4,𝔽2)12yes—
312C2×C4#47112yes—
313C2×C4#49511yescharacteristic
314C2×C4#49711yescharacteristic
315C8—44no—
316C8—82no—
317C8—82no—
318C8—161no—
319C22—22no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22#168012yes—
328C22#170012yes—
329C22#200011yescharacteristic
330C22#200511yescharacteristic
331C22#200811yescharacteristic
332C22#201511yescharacteristic
333C22#201911yescharacteristic
334C22#202011yescharacteristic
335C22#202111yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—82no—
344C4—84no—
345C4—161no—
346C4—161no—
347C4—161no—
348C4#160312yes—
349C4#160412yes—
350C2—22no—
351C2—22no—
352C2#991711yescharacteristic
353C2#995511yescharacteristic
354C2#998911yescharacteristic
355C2#1013911yescharacteristic
356C2#1017311yescharacteristic
357C2#1018011yescharacteristic
358C2#1027211yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ 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
#297572
#258562
C2×#20582
#98792
#83402
#76682
#70292