Finite Group Databasebeta

← #134489

Subgroups — #134489

Order 512.

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

normal subgroups only

This group has 604 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#14174C211yesmaximal, characteristic
2C2×#1354C212yesmaximal
3#10802C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7251C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C2×#395C2211yescharacteristic
8C2×#387—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13#1728—21no—
14C2×#324C2212yes—
15#1446C411yescharacteristic
16#1442C411yescharacteristic
17#1441C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20#1436—21no—
21#1434C411yescharacteristic
22#1354—24no—
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C22×M16—22no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2311yescharacteristic
34C22×C42—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40#395—21no—
41#395—21no—
42#395D412yes—
43#395Q812yes—
44#387—44no—
45C4×C4⋊C4—41no—
46C4×C4⋊C4—41no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#343—24no—
61#343—22no—
62#343—22no—
63#336—41no—
64#336—41no—
65#336—41no—
66#324—22no—
67#324—22no—
68C22×Q8—21no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8C4⋊C412yes—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×M16—24no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C22×C8—42no—
92C22×C8—42no—
93C4×Q8—44no—
94C4×Q8—44no—
95C4×Q8—44no—
96C4×Q8—44no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4C2×D411yescharacteristic
113C2×C4⋊C4C2×Q811yescharacteristic
114C2×C4⋊C4#4111yescharacteristic
115C2×C4⋊C4#4111yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—22no—
122C2×C42—21no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—42no—
129C4⋊C8—82no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#95—21no—
133#95—22no—
134#95—22no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95C4⋊C412yes—
147C24C2×C22⋊C411yescharacteristic
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—48no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
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—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4#9911yescharacteristic
216C22×C4#9911yescharacteristic
217C22×C4#10412yes—
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4#13512yes—
225C22×C4#13712yes—
226M16—44no—
227M16—44no—
228M16—82no—
229M16—82no—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—48no—
251C42—42no—
252C42—42no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23#38011yescharacteristic
259C23C2×#9911yescharacteristic
260C23#41011yescharacteristic
261C23#41611yescharacteristic
262C23#41911yescharacteristic
263C23C2×#10411yescharacteristic
264C23#42111yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—44no—
268Q8—44no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4AΣL(1,𝔽16)12yes—
342C2×C4#46412yes—
343C2×C4#47912yes—
344C8—84no—
345C8—162no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22#150011yescharacteristic
359C22#150811yescharacteristic
360C22#151511yescharacteristic
361C22#151611yescharacteristic
362C22#179911yescharacteristic
363C22#180511yescharacteristic
364C22#180611yescharacteristic
365C22C2×AΣL(1,𝔽16)11yescharacteristic
366C22#212011yescharacteristic
367C22#213311yescharacteristic
368C22#213511yescharacteristic
369C4—22no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—44no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4—81no—
383C4—161no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2#857411yescharacteristic
389C2#1179111yescharacteristic
390C2#1179211yescharacteristic
391C2#1184811yescharacteristic
392C2#1184911yescharacteristic
393C2#1279311yescharacteristic
394C2#1280411yescharacteristic
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
#141742
C2×#13542
#108022
#85582
#72512