Finite Group Databasebeta

← #497647

Subgroups — #497647

Order 512.

997 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36496C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13225C214yesmaximal, complemented
4#13223C212yesmaximal
5#12798C211yesmaximal, characteristic, complemented
6#12797C212yesmaximal, complemented
7C4×#354C211yesmaximal, characteristic, complemented
8C4×#353C212yesmaximal, complemented
9C2×#559C411yescharacteristic, complemented
10C2×#554C2211yescharacteristic
11#2861C412yescomplemented
12#2858C411yescharacteristic, complemented
13#2567C411yescharacteristic, complemented
14#2566C412yescomplemented
15#2544C2212yes—
16#2541C2211yescharacteristic
17#2461—24no—
18#2426—22no—
19#2361—22no—
20#2345—22no—
21C4×#127C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C411yescharacteristic, complemented
24C2×#354C2212yes—
25C2×#354C2212yes—
26C2×#353C2214yes—
27C2×#353C2214yes—
28#1996—22no—
29#1996—24no—
30#1799C2212yes—
31#1799C2214yes—
32C4×#99C2214yes—
33C4×#99C2212yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37#559—42no—
38#559—41no—
39#559—41no—
40#554—41no—
41#554—42no—
42#554—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#127C2312yes—
46C2×#125C2×C412yes—
47C2×#124—21no—
48C2×#124—22no—
49C2×#124C2×C412yes—
50C2×#124C2311yescharacteristic
51C2×#123—22no—
52C2×C22⋊Q8D412yes—
53C2×C22⋊Q8D414yes—
54C2×#121—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×D4—24no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2312yes—
62C2×C4×D4C2311yescharacteristic
63C22×C4⋊C4D412yes—
64C2×#99C2314yes—
65C2×#99C2312yes—
66C2×#99C2312yes—
67#396—24no—
68#395—22no—
69#394—24no—
70#392—22no—
71#391C2×C412yes—
72#391C2×C411yescharacteristic
73#390—22no—
74#390—24no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#390C2311yescharacteristic
79#389—22no—
80#389—42no—
81#388—24no—
82#388—24no—
83#388—22no—
84#387—44no—
85#386—24no—
86#385—24no—
87#384C2×C412yes—
88#382C2×C411yescharacteristic
89#380—24no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—42no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#354—24no—
102#354—42no—
103#353—28no—
104#353—44no—
105#342—22no—
106#342—24no—
107#342—22no—
108#342—24no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×D4—24no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C412yes—
116C22×D4C2411yescharacteristic
117C23×C4—22no—
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122#127—24no—
123#127—22no—
124#127—22no—
125#127—41no—
126#127—41no—
127#125—42no—
128#125—42no—
129#124—22no—
130#124—21no—
131#124—21no—
132#124—42no—
133#124—42no—
134#124—42no—
135#124—42no—
136#124—84no—
137#123—44no—
138#123—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—42no—
142C22⋊Q8—42no—
143#121—48no—
144C4×Q8—22no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—44no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—88no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22⋊C414yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C414yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4C2×D412yes—
189C2×C22⋊C4C2×D414yes—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—42no—
196C2×C42—44no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202#99—22no—
203#99—24no—
204#99—22no—
205#99—22no—
206#99—24no—
207#99#4114yes—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—24no—
213#95—42no—
214#95—44no—
215#95—44no—
216#95—42no—
217C24—22no—
218C24C22×D412yes—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—48no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—84no—
242C2×D4—82no—
243C2×D4C2×#4112yes—
244C2×D4H(5,𝔽2)11yescharacteristic
245C2×D42−1+411yescharacteristic
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C23×C411yescharacteristic
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—164no—
286C4⋊C4#9912yes—
287C4⋊C4#9911yescharacteristic
288C4⋊C4#9911yescharacteristic
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—84no—
307C42—22no—
308C42—22no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—21no—
314C42—22no—
315C42—41no—
316C42—44no—
317C42—42no—
318C42—44no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—84no—
323C42—164no—
324C23—22no—
325C23—22no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23C22×C22⋊C411yescharacteristic
334C23#54812yes—
335C23#55012yes—
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340Q8—162no—
341D4—42no—
342D4—41no—
343D4—41no—
344D4—82no—
345D4—82no—
346D4—82no—
347D4—88no—
348D4—88no—
349D4—82no—
350D4—162no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—84no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4C2×#9911yescharacteristic
393C2×C4C2×#9912yes—
394C2×C4C2×#9912yes—
395C2×C4C2×#9911yescharacteristic
396C2×C4C4×#4111yescharacteristic
397C2×C4#52011yescharacteristic
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—84no—
408C22—84no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22#234111yescharacteristic
413C22C22×#9911yescharacteristic
414C22#288411yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—44no—
422C4—42no—
423C4—42no—
424C4—44no—
425C4—41no—
426C4—81no—
427C4—84no—
428C4—81no—
429C4—81no—
430C4—162no—
431C4—164no—
432C4#212612yes—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—42no—
437C2—162no—
438C2#2253411yescharacteristic
439C2C2×#212611yescharacteristic
440C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#2541C22
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. #13225C2semidirect
5. #12798C2semidirect
6. #12797C2semidirect
7. C4×#354C2semidirect
8. C4×#353C2semidirect
9. C2×#559C4semidirect
11. #2861C4semidirect
12. #2858C4semidirect
13. #2567C4semidirect
14. #2566C4semidirect
23. C42×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#364962
#225132
#132252
#132232
#127982
#127972
C4×#3542
C4×#3532