Finite Group Databasebeta

← #134486

Subgroups — #134486

Order 512.

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

normal subgroups only

This group has 494 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#14242C211yesmaximal, characteristic
2C2×#1353C212yesmaximal
3#10802C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7251C212yesmaximal
6C2×#400C2211yescharacteristic
7C2×#389C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#343—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324C2212yes—
14#1445C411yescharacteristic
15#1442C411yescharacteristic
16#1441C2211yescharacteristic
17#1436—21no—
18#1436—21no—
19#1436—21no—
20#1434—21no—
21C4×#95C411yescharacteristic
22#1353—24no—
23#1280—21no—
24C22×M16—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4C2311yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36#400—21no—
37#400—21no—
38#400D412yes—
39#400Q812yes—
40#389—41no—
41#389—41no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58#343—24no—
59#343—22no—
60#343—22no—
61#336—41no—
62#336—41no—
63#336—41no—
64#324—22no—
65#324—22no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—42no—
84C22×C8—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4C4⋊C412yes—
100C2×C4⋊C4C2×D411yescharacteristic
101C2×C4⋊C4C2×Q811yescharacteristic
102C2×C4⋊C4#4111yescharacteristic
103C2×C4⋊C4#4111yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—81no—
115C2×C42—81no—
116C2×C42—81no—
117C4⋊C8—82no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120#95—21no—
121#95—22no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—81no—
135#95—81no—
136#95C4⋊C412yes—
137C24C2×C22⋊C411yescharacteristic
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
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×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4#9911yescharacteristic
197C22×C4#9911yescharacteristic
198C22×C4#10412yes—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4C22⋊Q811yescharacteristic
205C22×C4#13512yes—
206C22×C4#13712yes—
207M16—44no—
208M16—44no—
209M16—82no—
210M16—82no—
211C2×C8—44no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C42—42no—
228C42—42no—
229C42—42no—
230C42—81no—
231C42—81no—
232C42—81no—
233C42—84no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23#38011yescharacteristic
239C23C2×#9911yescharacteristic
240C23#41011yescharacteristic
241C23#41611yescharacteristic
242C23#41911yescharacteristic
243C23C2×#10411yescharacteristic
244C23#42111yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
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—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#35512yes—
318C2×C4#46412yes—
319C2×C4#47912yes—
320C8—84no—
321C8—162no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22#150011yescharacteristic
335C22#150811yescharacteristic
336C22#151511yescharacteristic
337C22#151611yescharacteristic
338C22#179911yescharacteristic
339C22#180511yescharacteristic
340C22#180611yescharacteristic
341C22#212111yescharacteristic
342C22#212311yescharacteristic
343C22C2×#35511yescharacteristic
344C22#213211yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—82no—
355C4—82no—
356C4—82no—
357C4—81no—
358C4—81no—
359C4—161no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2#857411yescharacteristic
365C2#1179311yescharacteristic
366C2#1179411yescharacteristic
367C2#1185211yescharacteristic
368C2#1185311yescharacteristic
369C2#1279611yescharacteristic
370C2#1280111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#142422
C2×#13532
#108022
#85582
#72512