Finite Group Databasebeta

← #497589

Subgroups — #497589

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13300C214yesmaximal, complemented
4#13299C212yesmaximal, complemented
5#12797C211yesmaximal, characteristic
6#12797C212yesmaximal
7C4×#353C212yesmaximal
8C4×#353C211yesmaximal, characteristic
9C2×#554C411yescharacteristic, complemented
10C2×#554C2211yescharacteristic
11#2868C411yescharacteristic, complemented
12#2859C412yescomplemented
13#2566C412yescomplemented
14#2566C2212yes—
15#2563C411yescharacteristic, complemented
16#2563C2211yescharacteristic
17#2538—22no—
18#2452—24no—
19D4×C4⋊C4—22no—
20#2345—22no—
21C4×#127C2211yescharacteristic
22C4×#127C2212yes—
23C42×Q8C411yescharacteristic, complemented
24C2×#353C2212yes—
25C2×#353C2211yescharacteristic
26C2×#353C2211yescharacteristic
27#1996—24no—
28#1996—22no—
29#1799C2212yes—
30#1799C2214yes—
31C4×#99C2212yes—
32C4×#99C2214yes—
33#1393C2214yes—
34#1393C2212yes—
35#1393C2214yes—
36#1393C2212yes—
37#554—41no—
38#554—42no—
39#554—41no—
40#554—42no—
41#554—41no—
42#554—41no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2311yescharacteristic
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8D414yes—
52C2×C22⋊Q8D412yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2311yescharacteristic
60C2×C4×D4C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D412yes—
63C2×#99C2314yes—
64C2×#99C2312yes—
65C2×#99C2312yes—
66#396—24no—
67#396—24no—
68#393—22no—
69#393—22no—
70#390—22no—
71#390—24no—
72#390C2×C412yes—
73#390C2×C411yescharacteristic
74#390C2312yes—
75#390C2311yescharacteristic
76#389—24no—
77#389—24no—
78#389—42no—
79#387—44no—
80#385—24no—
81#385—24no—
82#384—22no—
83#384C2×C411yescharacteristic
84#384C2×C412yes—
85#384C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#380—22no—
89#380—22no—
90#380—24no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—42no—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4C2×C412yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#353—24no—
102#353—22no—
103#353Q814yescomplemented
104#342—22no—
105#342—24no—
106#342—22no—
107#342—24no—
108C22×Q8—21no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×D4—22no—
112C22×D4C22×C411yescharacteristic
113C22×D4C2411yescharacteristic
114C23×C4—21no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—84no—
123#127—22no—
124#127C2×Q812yes—
125#127#4112yes—
126#125—42no—
127#125—42no—
128#125—42no—
129#125—42no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—44no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—48no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—88no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22⋊C414yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4C2×D412yes—
187C2×C22⋊C4C2×D414yes—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—42no—
193C2×C42—44no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199#99—22no—
200#99—24no—
201#99—22no—
202#99—24no—
203#99—22no—
204#99C2×Q814yes—
205#95—22no—
206#95—21no—
207#95—24no—
208#95—24no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—42no—
213C24—21no—
214C24C22×D412yes—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—21no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4C4×Q812yes—
232C2×D4C22×Q811yescharacteristic
233C2×D4C2×#4111yescharacteristic
234C2×D42−1+412yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C414yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C23×C411yescharacteristic
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—164no—
285C4⋊C4#9912yes—
286C4⋊C4#9911yescharacteristic
287C4⋊C4#9911yescharacteristic
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C42—24no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—42no—
310C42—44no—
311C42—44no—
312C42—42no—
313C42—84no—
314C42—164no—
315C42C4×Q812yes—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—41no—
324C23—41no—
325C23C22×C22⋊C411yescharacteristic
326C23#54812yes—
327C23D4×Q812yes—
328Q8—42no—
329Q8—42no—
330Q8—162no—
331Q8—162no—
332D4—41no—
333D4—41no—
334D4—42no—
335D4—42no—
336D4—48no—
337D4—48no—
338D4—42no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—162no—
380C2×C4—162no—
381C2×C4C2×#9912yes—
382C2×C4C2×#9912yes—
383C2×C4C2×#9911yescharacteristic
384C2×C4C2×#9911yescharacteristic
385C2×C4C2×C4×Q811yescharacteristic
386C2×C4#51911yescharacteristic
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22Q8×C22⋊C411yescharacteristic
402C22C22×#9911yescharacteristic
403C22#288411yescharacteristic
404C4—24no—
405C4—21no—
406C4—22no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—44no—
414C4—41no—
415C4—82no—
416C4—84no—
417C4—81no—
418C4—82no—
419C4—162no—
420C4—164no—
421C4#212612yes—
422C2—21no—
423C2—21no—
424C2—42no—
425C2—42no—
426C2—82no—
427C2Q8×#9911yescharacteristic
428C2C2×#212611yescharacteristic
429C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22513C2semidirect
3. #13300C2semidirect
4. #13299C2semidirect
9. C2×#554C4semidirect
11. #2868C4semidirect
12. #2859C4semidirect
13. #2566C4semidirect
15. #2563C4semidirect
23. C42×Q8C4semidirect
103. #353Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#225132
#133002
#132992
#127972
C4×#3532