Finite Group Databasebeta

← #102811

Subgroups — #102811

Order 512.

515 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 104 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#30187C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#27950C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic
5#9885C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9#2857C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2100—22no—
12#2100—21no—
13#2083—22no—
14#2072—21no—
15#2043—21no—
16#2024—21no—
17#1990—21no—
18#1990—21no—
19#1892—22no—
20#1852—22no—
21#1845—21no—
22#1834—21no—
23#1824—22no—
24C4×C22⋊C8C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—22no—
27C2×#336C2211yescharacteristic
28C2×#328—22no—
29C4⋊#95C2211yescharacteristic
30#1294—22no—
31C23×C8D412yes—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8D411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C42C2311yescharacteristic
41C2×#107—24no—
42C2×#103—22no—
43C2×#103—22no—
44C2×C22⋊C8D412yes—
45C2×C22⋊C8D412yes—
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48#391—21no—
49#391—21no—
50#389—21no—
51#389—21no—
52#389—21no—
53#389—21no—
54#384—21no—
55#384—21no—
56#384—21no—
57#384D411yescharacteristic
58#384D412yes—
59#384D411yescharacteristic
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64C2×#95—21no—
65C2×#95—21no—
66#340—22no—
67#340—24no—
68#340—24no—
69#340—24no—
70#340—22no—
71#340—44no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#328—44no—
77C22×Q8—21no—
78C22×Q8—21no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—42no—
90C22×C8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C22⋊Q8—42no—
94C22⋊Q8—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—82no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C42—42no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131#107—84no—
132#103—82no—
133#103—82no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8Q1614yes—
138C4×C8—42no—
139C4×C8—42no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—42no—
145#95—42no—
146C24#12711yescharacteristic
147C2×Q8—42no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8—81no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4#12011yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C2×Q1612yes—
184C22×C4#13512yes—
185C22×C4#13512yes—
186C22×C4AGL(1,ℤ/8)12yes—
187C22×C4#13712yes—
188C22×C4#13712yes—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—82no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—22no—
227C42—22no—
228C42—41no—
229C42—41no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23#39211yescharacteristic
235C23#49411yescharacteristic
236C23#49511yescharacteristic
237C23#49511yescharacteristic
238C23#49611yescharacteristic
239C23#49711yescharacteristic
240C23#49711yescharacteristic
241Q8—82no—
242Q8—82no—
243Q8—162no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—161no—
281C2×C4—161no—
282C2×C4—161no—
283C2×C4—162no—
284C2×C4—162no—
285C2×C4—161no—
286C2×C4#44912yes—
287C2×C4#45112yes—
288C2×C4#45212yes—
289C2×C4#45311yescharacteristic
290C2×C4#45411yescharacteristic
291C2×C4#45812yes—
292C2×C4#46212yes—
293C2×C4#46312yes—
294C2×C4#46412yes—
295C8—44no—
296C8—82no—
297C8—82no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22#171011yescharacteristic
311C22#171811yescharacteristic
312C22#171911yescharacteristic
313C22#172311yescharacteristic
314C22#200611yescharacteristic
315C22#200811yescharacteristic
316C22#200911yescharacteristic
317C22#201211yescharacteristic
318C22#201511yescharacteristic
319C22#201611yescharacteristic
320C22#202211yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—162no—
327C4—161no—
328C4—162no—
329C4—161no—
330C4—161no—
331C4—161no—
332C4#161512yes—
333C4#161812yes—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—21no—
338C2#995311yescharacteristic
339C2#995911yescharacteristic
340C2#996911yescharacteristic
341C2#1019311yescharacteristic
342C2#1020311yescharacteristic
343C2#1022311yescharacteristic
344C2#1031311yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#127
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
#301872
#279612
#279502
#113572
#98852
#96792
#70162