Finite Group Databasebeta

← #100688

Subgroups — #100688

Order 512.

389 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 90 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#29970C211yesmaximal, characteristic
2#21958C211yesmaximal, characteristic
3#7727C211yesmaximal, characteristic
4#7668C211yesmaximal, characteristic
5#7579C211yesmaximal, characteristic
6#7044C211yesmaximal, characteristic
7#7020C211yesmaximal, characteristic
8C2×#433—21no—
9#2857C2211yescharacteristic
10#2296C2211yescharacteristic
11#2100C411yescharacteristic
12#2100C411yescharacteristic
13#2088—21no—
14#2086—21no—
15#2058C411yescharacteristic
16#2057C411yescharacteristic
17#1940—21no—
18#1940—21no—
19#1939—21no—
20#1939—21no—
21#1854—21no—
22#1825—21no—
23#1817C2211yescharacteristic
24#1798—21no—
25#1797C2211yescharacteristic
26#1797C2211yescharacteristic
27#1797C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1272—21no—
30#1272—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1271—21no—
35C23×C8—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×C22⋊Q8—21no—
38C2×#117D411yescharacteristic
39C2×#117D411yescharacteristic
40C22×C42C2311yescharacteristic
41#433—42no—
42#433—42no—
43C2×#107—21no—
44C2×#106—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×#97—21no—
57C2×C4×C8—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389C2×C411yescharacteristic
62#389C2×C411yescharacteristic
63#389D411yescharacteristic
64#389D411yescharacteristic
65#385—21no—
66#385—21no—
67#384C2×C411yescharacteristic
68#380—21no—
69#380—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—81no—
105C22×C8—81no—
106C4⋊Q8—81no—
107C22⋊Q8—81no—
108C22⋊Q8—81no—
109#117—21no—
110#117—21no—
111#117—81no—
112#117SD1612yes—
113#117Q1612yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—41no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—81no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138#107—81no—
139#106—81no—
140C4⋊C8—42no—
141C4⋊C8—42no—
142C4⋊C8—81no—
143C4⋊C8—81no—
144C4⋊C8—81no—
145C4⋊C8—81no—
146C4⋊C8—81no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—81no—
154#97—42no—
155C4×C8—42no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—41no—
161#95—41no—
162C24#12311yescharacteristic
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—81no—
166C2×Q8—81no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C2×SD1611yescharacteristic
190C22×C4C2×Q1611yescharacteristic
191C22×C4AGL(1,ℤ/8)11yescharacteristic
192C22×C4#13711yescharacteristic
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—81no—
213C2×C8—82no—
214C2×C8—81no—
215C2×C8—82no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4#10312yes—
238C4⋊C4#10312yes—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—81no—
242C22⋊C4—82no—
243C42—22no—
244C42—22no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#38611yescharacteristic
258C23#48311yescharacteristic
259C23#48411yescharacteristic
260Q8—161no—
261Q8—161no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4C2×#10311yescharacteristic
304C2×C4#41711yescharacteristic
305C2×C4#44311yescharacteristic
306C2×C4#44411yescharacteristic
307C2×C4#45011yescharacteristic
308C2×C4#45111yescharacteristic
309C2×C4#47111yescharacteristic
310C2×C4#47211yescharacteristic
311C8—81no—
312C8—81no—
313C8—81no—
314C8—81no—
315C8—81no—
316C8—81no—
317C8—82no—
318C8—161no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#187211yescharacteristic
330C22#187611yescharacteristic
331C22#188111yescharacteristic
332C22#190211yescharacteristic
333C22#190711yescharacteristic
334C22#191211yescharacteristic
335C22#191311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—81no—
349C4—81no—
350C4—161no—
351C4—161no—
352C2—41no—
353C2—41no—
354C2#899311yescharacteristic
355C2#916711yescharacteristic
356C2#920011yescharacteristic
357C2#926411yescharacteristic
358C2#928611yescharacteristic
359C2#933111yescharacteristic
360C2#937211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×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
#299702
#219582
#77272
#76682
#75792
#70442
#70202