Finite Group Databasebeta

← #100545

Subgroups — #100545

Order 512.

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

normal subgroups only

This group has 394 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#33154C211yesmaximal, characteristic
2#20774C211yesmaximal, characteristic
3#8708C211yesmaximal, characteristic
4#7688C211yesmaximal, characteristic
5#7667C211yesmaximal, characteristic
6#7045C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×#117C2211yescharacteristic
10#1988C411yescharacteristic
11#1988C411yescharacteristic
12#1948C411yescharacteristic
13#1947C411yescharacteristic
14#1921—21no—
15#1920—21no—
16#1854—21no—
17#1840—22no—
18#1839—22no—
19#1825—21no—
20#1797C2211yescharacteristic
21#1797C2211yescharacteristic
22C4×C22⋊C8C2211yescharacteristic
23#1729—21no—
24#1729—21no—
25#1728C2211yescharacteristic
26C2×#336C2211yescharacteristic
27#1279—21no—
28#1279—21no—
29#1272—22no—
30#1271—22no—
31C22×M16—21no—
32C23×C8—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×#117D411yescharacteristic
36C2×#117D411yescharacteristic
37C22×C42C2311yescharacteristic
38C2×#107—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C4×C8—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#384C2×C411yescharacteristic
51#380—21no—
52#380—21no—
53C4×C4⋊C4C2×C411yescharacteristic
54C4×C4⋊C4C2×C411yescharacteristic
55C4×C4⋊C4D411yescharacteristic
56C4×C4⋊C4D411yescharacteristic
57C4×C22⋊C4—21no—
58C4×C22⋊C4—21no—
59#376—21no—
60C43—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#336—24no—
68#336—24no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336D412yes—
75#336D412yes—
76C22×Q8—21no—
77C23×C4C2×D411yescharacteristic
78C23×C4#4111yescharacteristic
79C23×C4#4111yescharacteristic
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C22⋊Q8—82no—
101#117—21no—
102#117—21no—
103#117—21no—
104#117—21no—
105#117—41no—
106#117—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—41no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132#107—42no—
133#106—42no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—81no—
143C4×C8—41no—
144C4×C8—41no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—41no—
150#95—41no—
151C24#12311yescharacteristic
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—81no—
155C2×Q8—81no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#13511yescharacteristic
181C22×C4#13511yescharacteristic
182C22×C4#13711yescharacteristic
183C22×C4#13711yescharacteristic
184M16—81no—
185M16—81no—
186M16—81no—
187M16—81no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
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—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C22⋊C4—42no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C42—42no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—81no—
244C42—81no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23#38611yescharacteristic
252C23#48511yescharacteristic
253C23#48511yescharacteristic
254Q8—161no—
255Q8—161no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
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—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#41011yescharacteristic
299C2×C4#41011yescharacteristic
300C2×C4#41611yescharacteristic
301C2×C4#41911yescharacteristic
302C2×C4#42111yescharacteristic
303C2×C4#42111yescharacteristic
304C2×C4#44311yescharacteristic
305C2×C4#44411yescharacteristic
306C2×C4#45211yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#47111yescharacteristic
309C2×C4#47311yescharacteristic
310C8—82no—
311C8—82no—
312C8—82no—
313C8—81no—
314C8—81no—
315C8—161no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22#187311yescharacteristic
327C22#187611yescharacteristic
328C22#188111yescharacteristic
329C22#189711yescharacteristic
330C22#189911yescharacteristic
331C22#190311yescharacteristic
332C22#190811yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—161no—
346C4—161no—
347C4#151511yescharacteristic
348C4#151511yescharacteristic
349C4#152111yescharacteristic
350C4#152111yescharacteristic
351C2—41no—
352C2—41no—
353C2#899411yescharacteristic
354C2#915311yescharacteristic
355C2#917011yescharacteristic
356C2#921411yescharacteristic
357C2#924211yescharacteristic
358C2#931311yescharacteristic
359C2#934611yescharacteristic
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
#331542
#207742
#87082
#76882
#76672
#70452
#70292