Finite Group Databasebeta

← #115771

Subgroups — #115771

Order 512.

832 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 142 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#21955C211yesmaximal, characteristic, complemented
2C2×#1533C212yesmaximal
3#14153C211yesmaximal, characteristic
4C2×#1285C212yesmaximal, complemented
5#7912C211yesmaximal, characteristic, complemented
6C22×#124C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×#391C412yes—
10C2×#390C412yes—
11C2×C4×C4⋊C4C2211yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13#1940—24no—
14#1797—22no—
15#1533—24no—
16#1533—24no—
17C2×#329—24no—
18#1439—21no—
19#1439—21no—
20#1437—22no—
21#1285—24no—
22#1280—21no—
23#1280—21no—
24C23×Q8C2×C411yescharacteristic
25C23×D4C2×C411yescharacteristic
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—21no—
29C2×#124—21no—
30C2×#124—22no—
31C2×#124—22no—
32C2×#124D412yes—
33C2×#124Q812yes—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C412yes—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C42C2×C412yes—
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432—22no—
48#432—22no—
49#432—22no—
50#432D414yes—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—24no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×#97—22no—
57C2×#97D414yes—
58#391—44no—
59#390—44no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95D411yescharacteristic
68C2×#95D411yescharacteristic
69#336—24no—
70#329—216no—
71C25—21no—
72C22×Q8—21no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8C4⋊C412yes—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—42no—
82C22×D4—42no—
83C22×D4C4⋊C412yes—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—24no—
92C22×C8—42no—
93#124—24no—
94#124—24no—
95#124—24no—
96#124—24no—
97#124—42no—
98#124—42no—
99#124—42no—
100#124—42no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—82no—
121C2×C22⋊C4—82no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—82no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×Q811yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4112yes—
134C2×C42#4112yes—
135C2×C42#4111yescharacteristic
136C4⋊C8—24no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8—84no—
141C22⋊C8—24no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145#97—24no—
146#97—24no—
147#97—24no—
148#97—42no—
149#95—22no—
150#95—22no—
151#95—44no—
152#95—44no—
153#95—41no—
154#95—41no—
155C24—22no—
156C24—42no—
157C24—42no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—42no—
165C24—42no—
166C24C2×C22⋊C411yescharacteristic
167C2×Q8—24no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—84no—
172C2×Q8—82no—
173C2×Q8—82no—
174C2×D4—24no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—84no—
179C2×D4—82no—
180C2×D4—82no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4#10014yes—
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4#11711yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4#12011yescharacteristic
213C22×C4#12112yes—
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12112yes—
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12711yescharacteristic
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—84no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—88no—
236C22⋊C4—88no—
237C42—44no—
238C42—82no—
239C42—84no—
240C42#12012yes—
241C42#12112yes—
242C42C22⋊Q812yes—
243C42C22⋊Q812yes—
244C42#12312yes—
245C42#12312yes—
246C42#12412yes—
247C42#12612yes—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—44no—
252C23—42no—
253C23—42no—
254C23—44no—
255C23—42no—
256C23—82no—
257C23—82no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—82no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23#38011yescharacteristic
275C23#38611yescharacteristic
276C23#39011yescharacteristic
277C23C2×#10012yes—
278C23#41312yes—
279Q8—84no—
280Q8—84no—
281D4—84no—
282D4—84no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4#38512yes—
323C2×C4#38812yes—
324C2×C4#39211yescharacteristic
325C2×C4#39311yescharacteristic
326C2×C4#39412yes—
327C2×C4#39612yes—
328C2×C4#39711yescharacteristic
329C2×C4#39911yescharacteristic
330C8—84no—
331C8—84no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—84no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—161no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22—162no—
350C22—162no—
351C22#143911yescharacteristic
352C22#180012yes—
353C22#188412yes—
354C22#190012yes—
355C22#191111yescharacteristic
356C22#191211yescharacteristic
357C22#196511yescharacteristic
358C22#196711yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—84no—
369C4—82no—
370C4—82no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—162no—
375C2#944712yes—
376C2#944812yes—
377C2#1104411yescharacteristic
378C2#1105611yescharacteristic
379C2#1107211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21955C2semidirect
4. C2×#1285C2semidirect
5. #7912C2semidirect

Maximal subgroups

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

subgroupindex
#219552
C2×#15332
#141532
C2×#12852
#79122