Finite Group Databasebeta

← #99372

Subgroups — #99372

Order 512.

833 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 833 conjugacy classes of subgroups — too many to draw. Only its 154 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#14440C211yesmaximal, characteristic
2#11267C212yesmaximal, complemented
3#11049C211yesmaximal, characteristic
4C2×#1270C211yesmaximal, characteristic
5#7025C212yesmaximal, complemented
6C22×#102—21no—
7C2×#390C2211yescharacteristic
8C2×#386—21no—
9C2×#385—22no—
10C2×#384C2211yescharacteristic
11C2×#340—22no—
12C2×#340—22no—
13C2×#340—21no—
14C2×#336C2211yescharacteristic
15C2×#336C2212yes—
16#1441—21no—
17#1439—22no—
18C4⋊#95C2212yes—
19#1270C414yescomplemented
20C23×D4D411yescharacteristic
21C24×C4—21no—
22C23×C8—21no—
23C23×C8—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C42C2311yescharacteristic
34C2×#102—42no—
35C2×#102—42no—
36C2×#102—42no—
37#390—22no—
38#390C2×C412yes—
39#390Q812yes—
40#386—42no—
41#386—42no—
42#385—44no—
43#385—44no—
44#384—42no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—21no—
49#340—44no—
50#340—44no—
51#340—42no—
52#336—22no—
53#336—22no—
54#336C2×C414yescomplemented
55C25—21no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—41no—
60C22×D4—41no—
61C22×D4—41no—
62C22×D4—41no—
63C22×D4C4⋊C412yes—
64C22×D4D812yes—
65C22×D4Q1612yes—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—44no—
70C23×C4—42no—
71C23×C4—41no—
72C23×C4—81no—
73C23×C4—82no—
74C23×C4—81no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—41no—
82C22×C8—42no—
83C22×C8—41no—
84C22×C8—42no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—41no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—81no—
115C2×C4⋊C4C4⋊C412yes—
116C2×C4⋊C4SD1614yes—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—82no—
127C2×C22⋊C4—81no—
128C2×C22⋊C4—81no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×Q811yescharacteristic
133C2×C42#4112yes—
134#102—48no—
135#95—44no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—81no—
141C24—22no—
142C24—22no—
143C24—22no—
144C24—42no—
145C24—42no—
146C24—41no—
147C24—42no—
148C24—42no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24#12011yescharacteristic
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4#10714yes—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—84no—
198C22×C4—88no—
199C22×C4—88no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4—81no—
203C22×C4—82no—
204C22×C4—81no—
205C22×C4—82no—
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4C4×D412yes—
208C22×C4C22⋊Q812yes—
209C22×C4#12311yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×SD1612yes—
212C22×C4C2×Q1611yescharacteristic
213C22×C4#13512yes—
214C22×C4AGL(1,ℤ/8)11yescharacteristic
215C22×C4AGL(1,ℤ/8)12yes—
216C22×C4AGL(1,ℤ/8)12yes—
217C22×C4#13711yescharacteristic
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—81no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—81no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—88no—
252C22⋊C4—84no—
253C42—41no—
254C42—41no—
255C42—42no—
256C23—24no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—44no—
261C23—42no—
262C23—44no—
263C23—42no—
264C23—42no—
265C23—44no—
266C23—44no—
267C23—44no—
268C23—44no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—82no—
273C23—81no—
274C23—81no—
275C23—82no—
276C23—81no—
277C23—81no—
278C23—82no—
279C23#38511yescharacteristic
280C23#44711yescharacteristic
281C23#44912yes—
282C23#45012yes—
283C23#45111yescharacteristic
284C23#45311yescharacteristic
285C23#45412yes—
286C23#45511yescharacteristic
287C23#45512yes—
288C23B(4,𝔽2)11yescharacteristic
289C23#45811yescharacteristic
290D4—44no—
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—816no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—161no—
341C2×C4—162no—
342C2×C4—161no—
343C2×C4C2×#10711yescharacteristic
344C2×C4#42811yescharacteristic
345C2×C4C4×SD1612yes—
346C2×C4#44212yes—
347C2×C4#47612yes—
348C2×C4#47812yes—
349C2×C4#48011yescharacteristic
350C2×C4#48411yescharacteristic
351C8—82no—
352C8—84no—
353C22—44no—
354C22—48no—
355C22—82no—
356C22—82no—
357C22—84no—
358C22—84no—
359C22—82no—
360C22—82no—
361C22—84no—
362C22—82no—
363C22—82no—
364C22#159511yescharacteristic
365C22#159712yes—
366C22#160112yes—
367C22#160211yescharacteristic
368C22#160711yescharacteristic
369C22#161212yes—
370C22#161411yescharacteristic
371C22#161612yes—
372C22#162011yescharacteristic
373C22#162512yes—
374C22#162911yescharacteristic
375C22#163712yes—
376C22#163912yes—
377C22#164212yes—
378C22#164412yes—
379C22#165711yescharacteristic
380C22#165911yescharacteristic
381C22#166212yes—
382C22#185211yescharacteristic
383C22#185611yescharacteristic
384C22#186012yes—
385C22#186412yes—
386C22#186711yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—84no—
396C4—82no—
397C4—162no—
398C4—162no—
399C2—84no—
400C2#773611yescharacteristic
401C2#775011yescharacteristic
402C2#777412yes—
403C2#779512yes—
404C2#782111yescharacteristic
405C2#782611yescharacteristic
406C2#885311yescharacteristic
407C2#887012yes—
408C2#889012yes—
409C2#891411yescharacteristic
410C2#892411yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11267C2semidirect
5. #7025C2semidirect
19. #1270C4semidirect
54. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
#144402
#112672
#110492
C2×#12702
#70252