Finite Group Databasebeta

← #627739

Subgroups — #627739

Order 512.

4100 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#2858C211yesmaximal, characteristic, complemented
3C2×#2641C211yesmaximal, characteristic, complemented
4C2×#2619C211yesmaximal, characteristic
5#15909C218yesmaximal, complemented
6#15882C218yesmaximal, complemented
7#15215C214yesmaximal, complemented
8#14709C211yesmaximal, characteristic, complemented
9#14274C211yesmaximal, characteristic, complemented
10#14026C214yesmaximal, complemented
11#13761C211yesmaximal, characteristic, complemented
12C2×#541—28no—
13C2×#541—28no—
14C2×#541—28no—
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#126C2214yes—
18C22×#125C2214yes—
19C22×#125C2211yescharacteristic
20C22×#124C2214yes—
21C22×C22⋊Q8C2212yes—
22C22×#121C2212yes—
23C22×C4×Q8C2212yes—
24C22×#117C2211yescharacteristic
25#2858—28no—
26#2641—28no—
27#2619—28no—
28C2×#394C2214yes—
29C2×#394C2218yes—
30C2×#392C2214yes—
31C2×#391C2214yes—
32C2×#390C2218yes—
33C2×#390C2212yes—
34C2×#390C2214yes—
35C2×#382C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×#376C2211yescharacteristic
38C2×C43C2211yescharacteristic
39#1445C2218yes—
40#1439C22116yes—
41#1439C22116yes—
42#1438C22116yes—
43#1437C22116yes—
44#1437C2214yes—
45#1436C2214yes—
46#1436C2218yes—
47C4⋊#95C2214yes—
48C4×#95C2212yes—
49C23×Q8C2312yes—
50C23×D4C2312yes—
51C23×D4C2314yes—
52C23×D4C2314yes—
53C24×C4C2311yescharacteristic
54#541—2128no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8D418yescomplemented
58C2×#127—24no—
59C2×#127—24no—
60C2×#127—24no—
61C2×#126—216no—
62C2×#126—216no—
63C2×#126—216no—
64C2×#125—216no—
65C2×#125—216no—
66C2×#125—216no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125D418yescomplemented
70C2×#124—216no—
71C2×#124—216no—
72C2×#124—48no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×#121—28no—
77C2×#121—48no—
78C2×C4×Q8—28no—
79C2×C4×Q8—28no—
80C2×C4×Q8—28no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117D418yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C23116yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2318yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C23116yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2314yes—
103C22×C42C2311yescharacteristic
104#394—216no—
105#394—232no—
106#392—216no—
107#391—48no—
108#390—232no—
109#390—216no—
110#390—28no—
111#382—24no—
112C4×C22⋊C4—28no—
113#376—24no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C23116yes—
119C2×#95C23116yes—
120C43—24no—
121C25C2414yes—
122C25C2411yescharacteristic
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—28no—
127C22×Q8C2×D418yes—
128C22×D4—28no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—28no—
132C22×D4—24no—
133C22×D4—28no—
134C22×D4—28no—
135C22×D4—28no—
136C22×D4—28no—
137C22×D4—48no—
138C22×D4—416no—
139C22×D4—416no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4C2×D414yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2412yes—
150C23×C4C2411yescharacteristic
151C23×C4C2414yes—
152C23×C4C2412yes—
153C23×C4C2411yescharacteristic
154C4⋊Q8—232no—
155#127—416no—
156#126—2128no—
157#125—2128no—
158#125—232no—
159#124—464no—
160C22⋊Q8—264no—
161#121—432no—
162C4×Q8—264no—
163#117—232no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4C2×D418yes—
181C2×C4⋊C4C2×D418yes—
182C2×C4⋊C4C2×D4116yes—
183C2×C4⋊C4C2×D4116yes—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—232no—
191C2×C22⋊C4—232no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—216no—
194C2×C22⋊C4—232no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4—48no—
197C2×C22⋊C4—416no—
198C2×C22⋊C4—416no—
199C2×C22⋊C4—416no—
200C2×C22⋊C4—416no—
201C2×C22⋊C4C2×D418yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—28no—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—48no—
214C2×C42C2×D414yes—
215C2×C42C2×D414yes—
216C2×C42C2×D414yes—
217C2×C42C2×D414yes—
218C2×C42#4118yes—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—416no—
223#95—48no—
224#95—416no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—28no—
230C24—24no—
231C24—24no—
232C24—24no—
233C24—28no—
234C24—28no—
235C24—216no—
236C24—216no—
237C24C22×D412yes—
238C24C2511yescharacteristic
239C2×Q8—216no—
240C2×Q8—232no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—416no—
244C2×D4—432no—
245C2×D4—416no—
246C2×D4—416no—
247C2×D4—48no—
248C2×D4—48no—
249C2×D4—48no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—816no—
253C2×D4—816no—
254C2×D4—816no—
255C2×D4—832no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—216no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—216no—
285C22×C4—28no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4C22×D414yes—
289C22×C4C22×D418yes—
290C22×C4C22×D418yes—
291C22×C4C22×D412yes—
292C22×C4C22×D414yes—
293C22×C4C22×D414yes—
294C22×C4C22×D414yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C2×#4118yes—
299C22×C4C2×#4114yes—
300C22×C4H(5,𝔽2)14yes—
301C4⋊C4—232no—
302C4⋊C4—232no—
303C4⋊C4—216no—
304C4⋊C4—216no—
305C4⋊C4—232no—
306C4⋊C4—232no—
307C4⋊C4—48no—
308C22⋊C4—232no—
309C22⋊C4—216no—
310C22⋊C4—464no—
311C22⋊C4—416no—
312C22⋊C4—416no—
313C22⋊C4—832no—
314C22⋊C4—816no—
315C42—232no—
316C42—28no—
317C42—216no—
318C42—416no—
319C42—48no—
320C42#120116yes—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—28no—
325C23—44no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—48no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—416no—
336C23—48no—
337C23—48no—
338C23—416no—
339C23—48no—
340C23—416no—
341C23—48no—
342C23—416no—
343C23—416no—
344C23—416no—
345C23C23×D411yescharacteristic
346C23C23×D411yescharacteristic
347C23C23×D411yescharacteristic
348C23C22×#4112yes—
349C23C2×H(5,𝔽2)12yes—
350C23#58518yes—
351Q8—432no—
352D4—832no—
353D4—832no—
354D4—816no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—216no—
367C2×C4—28no—
368C2×C4—216no—
369C2×C4—28no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—416no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—416no—
390C2×C4C2×#12018yes—
391C2×C4C2×#12014yes—
392C2×C4#53214yes—
393C2×C4#53514yes—
394C2×C4#54618yes—
395C2×C4#55018yes—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—42no—
400C22—84no—
401C22—84no—
402C22—88no—
403C22—84no—
404C22—84no—
405C22—816no—
406C22—88no—
407C22—84no—
408C22—816no—
409C22C22×#12011yescharacteristic
410C22C2×#53211yescharacteristic
411C22C2×#53511yescharacteristic
412C22#345514yes—
413C22C2×#54612yes—
414C22C2×#55012yes—
415C22#348714yes—
416C22#350014yes—
417C22#350014yes—
418C22#351214yes—
419C22#356218yes—
420C4—28no—
421C4—48no—
422C4—44no—
423C4—48no—
424C4—42no—
425C4—48no—
426C4—84no—
427C4—84no—
428C4#243118yes—
429C2—82no—
430C2—164no—
431C2C2×#243111yescharacteristic
432C2#3533212yes—
433C2#3567612yes—
434C2#3742412yes—
435C2#3770114yes—
436C2#3938214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#541C2semidirect
2. C2×#2858C2semidirect
3. C2×#2641C2semidirect
5. #15909C2semidirect
6. #15882C2semidirect
7. #15215C2semidirect
8. #14709C2semidirect
9. #14274C2semidirect
10. #14026C2semidirect
11. #13761C2semidirect
57. C2×C4⋊Q8D4semidirect
69. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#28582
C2×#26412
C2×#26192
#159092
#158822
#152152
#147092
#142742
#140262
#137612