Finite Group Databasebeta

← #496394

Subgroups — #496394

Order 512.

1148 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C2×#2566C211yesmaximal, characteristic
2#20722C212yesmaximal
3#13300C218yesmaximal, complemented
4#12856C212yesmaximal
5#12856C212yesmaximal
6C22×#125C411yescharacteristic, complemented
7C22×C4×D4C2211yescharacteristic
8#2566—22no—
9#2566—22no—
10#2566C414yescomplemented
11#2566C2214yes—
12C2×#396—22no—
13C2×#390C2211yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#385—22no—
16C2×#384C411yescharacteristic, complemented
17C2×#382C411yescharacteristic, complemented
18#2280—22no—
19C2×C4×C4⋊C4C411yescharacteristic, complemented
20#1819—22no—
21#1799—22no—
22#1799C2214yes—
23#1781C2214yes—
24C4×#99—22no—
25C4×#99C2214yes—
26#1393C2218yes—
27#1393C2218yes—
28C23×D4C2311yescharacteristic
29C24×C4D412yes—
30C2×#125—22no—
31C2×#125—42no—
32C2×#125—41no—
33C2×#125—41no—
34C2×#125C2×C414yes—
35C2×C4×D4—42no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4C2×C412yes—
39C2×C4×D4D412yes—
40C2×C4×D4D412yes—
41C2×C4×D4C2312yes—
42C2×#117—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D412yes—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C2×#99—24no—
57C2×#99C2318yes—
58#396—24no—
59#396—24no—
60#396—24no—
61#396—24no—
62#390—22no—
63#390—22no—
64#390C2×C412yes—
65#390C2×C412yes—
66#390C2312yes—
67#390C2312yes—
68#385—24no—
69#385—24no—
70#385—24no—
71#385—24no—
72#384—21no—
73#384—21no—
74#384C2×C414yes—
75#382—21no—
76#382—21no—
77#382C2×C414yes—
78#381—42no—
79#380—24no—
80#380—42no—
81#379—24no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4C2×C414yes—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—42no—
86C2×#95—22no—
87C2×#95—21no—
88#342—24no—
89#342—24no—
90C25C2×D412yes—
91C22×D4—24no—
92C22×D4—42no—
93C22×D4—41no—
94C22×D4—41no—
95C22×D4C22×C411yescharacteristic
96C22×D4C2×D412yes—
97C22×D4C2411yescharacteristic
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—84no—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115#125—44no—
116#125—44no—
117#125—44no—
118#125—44no—
119C4×D4—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127#117—88no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—44no—
177C2×C42C22⋊C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181#99—216no—
182#99—48no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—44no—
187#95—44no—
188C24—24no—
189C24—22no—
190C24—22no—
191C24—42no—
192C24—44no—
193C24—44no—
194C24C2×C22⋊C411yescharacteristic
195C24C2×C22⋊C412yes—
196C24C22×D412yes—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—48no—
206C2×D4—48no—
207C2×D42−1+414yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—48no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—48no—
245C22×C4—84no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4#9912yes—
253C22×C4#9911yescharacteristic
254C22×C4#9911yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C23×C411yescharacteristic
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—48no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C4⋊C4—88no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—84no—
284C22⋊C4—84no—
285C22⋊C4—88no—
286C22⋊C4—88no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—88no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—44no—
306C23—44no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—416no—
312C23—44no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—88no—
317C23C2×#9911yescharacteristic
318C23C2×#9911yescharacteristic
319C23C22×C22⋊C411yescharacteristic
320C23#53614yes—
321D4—44no—
322D4—48no—
323D4—44no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—88no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4C2×#9912yes—
372C2×C4C2×#9912yes—
373C2×C4#41012yes—
374C2×C4#51912yes—
375C22—24no—
376C22—24no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—24no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—88no—
393C22—88no—
394C22—82no—
395C22#212612yes—
396C22#212712yes—
397C22#231011yescharacteristic
398C22C22×#9911yescharacteristic
399C22C2×#41011yescharacteristic
400C4—22no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4—168no—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—22no—
416C2—44no—
417C2—82no—
418C2#2182511yescharacteristic
419C2C2×#212611yescharacteristic
420C2C2×#212711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13300C2semidirect
6. C22×#125C4semidirect
10. #2566C4semidirect
16. C2×#384C4semidirect
17. C2×#382C4semidirect
19. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
#207222
#133002
#128562