Finite Group Databasebeta

← #134333

Subgroups — #134333

Order 512.

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

normal subgroups only

This group has 524 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#14000C211yesmaximal, characteristic
2C2×#1319C212yesmaximal
3#8560C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7189C212yesmaximal
6C2×#391C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×#384C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12#1797—21no—
13C2×#324C2212yes—
14C2×#324C2212yes—
15#1442—21no—
16#1442—21no—
17#1436—21no—
18#1436—21no—
19C4⋊#95C2211yescharacteristic
20C4×#95C411yescharacteristic
21#1319—24no—
22#1280—21no—
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39#391—21no—
40#391—21no—
41#391—41no—
42#391—41no—
43#391—42no—
44#391D412yes—
45#391Q812yes—
46#384—42no—
47C4×C4⋊C4—41no—
48C4×C4⋊C4—41no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61#336—41no—
62#336—41no—
63#324—22no—
64#324—22no—
65#324—22no—
66#324—22no—
67C22×Q8—21no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C4⋊C412yes—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4C2×D411yescharacteristic
103C2×C4⋊C4C2×Q811yescharacteristic
104C2×C4⋊C4#4111yescharacteristic
105C2×C4⋊C4#4111yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—81no—
121C2×C42—81no—
122C2×C42C4⋊C412yes—
123C4⋊C8—82no—
124C4⋊C8—82no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—24no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—81no—
143#95—81no—
144C24C2×C22⋊C411yescharacteristic
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—82no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
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—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#9911yescharacteristic
214C22×C4#9911yescharacteristic
215C22×C4#10012yes—
216C22×C4C2×C4⋊C411yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4#13112yes—
223C22×C4#13112yes—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—44no—
235C4⋊C4—82no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42—81no—
245C42—81no—
246C42—82no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23#38011yescharacteristic
252C23#40811yescharacteristic
253C23#40811yescharacteristic
254C23C2×#9911yescharacteristic
255C23#41011yescharacteristic
256C23C2×#10011yescharacteristic
257C23#41311yescharacteristic
258Q8—84no—
259Q8—84no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
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—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
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#35612yes—
333C2×C4#42412yes—
334C2×C4#43312yes—
335C8—162no—
336C8—162no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22#146811yescharacteristic
350C22#146911yescharacteristic
351C22#147111yescharacteristic
352C22#147311yescharacteristic
353C22#179811yescharacteristic
354C22#179911yescharacteristic
355C22#180011yescharacteristic
356C22#212711yescharacteristic
357C22#212911yescharacteristic
358C22C2×#35611yescharacteristic
359C22#213211yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—44no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—42no—
370C4—82no—
371C4—81no—
372C4—81no—
373C4—82no—
374C4—161no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2#856111yescharacteristic
380C2#1155511yescharacteristic
381C2#1155611yescharacteristic
382C2#1156611yescharacteristic
383C2#1156811yescharacteristic
384C2#1280011yescharacteristic
385C2#1280211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
#140002
C2×#13192
#85602
#85582
#71892