Finite Group Databasebeta

← #380758

Subgroups — #380758

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic
5C2×#1746C211yesmaximal, characteristic
6#14283C211yesmaximal, characteristic
7#14282C211yesmaximal, characteristic
8#11379C212yesmaximal
9#11378C212yesmaximal
10#11378C212yesmaximal
11#11377C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1991—24no—
25#1990—24no—
26#1836—24no—
27#1746—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1446—22no—
39#1445—22no—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#501—232no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107C2×C418yes—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106C2×C418yes—
82C2×#97C2×C414yes—
83C2×#97D414yes—
84C2×#97D414yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8D414yes—
88#384—22no—
89#384—22no—
90#382—22no—
91#382—22no—
92#376—22no—
93#376—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95D412yes—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—24no—
111#336—24no—
112C22×Q8—24no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—82no—
116C22×Q8—82no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C2×D414yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C22×C414yes—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×D414yes—
143C22×C8C2×D414yes—
144C4⋊Q8—28no—
145C4⋊Q8—28no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148#125—28no—
149#125—28no—
150#125—44no—
151#125—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C414yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207#107—216no—
208#106—216no—
209#97C22⋊C4116yes—
210C4×C8—28no—
211#95—24no—
212#95—24no—
213#95—42no—
214#95—42no—
215#95—84no—
216#95—84no—
217#95—84no—
218#95—84no—
219C24C22×D411yescharacteristic
220C2×Q8—48no—
221C2×Q8—88no—
222C2×Q8—84no—
223C2×Q8—84no—
224C2×Q8—88no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4#12712yes—
268C22×C4#12712yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C42−1+412yes—
276C22×C42−1+412yes—
277C22×C42−1+412yes—
278C22×C42−1+412yes—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8C2×C22⋊C418yes—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42C2×C22⋊C414yes—
327C42C4×D414yes—
328C42#12114yes—
329C42#12414yes—
330C23C2×C4×D411yescharacteristic
331C23C2×#12111yescharacteristic
332C23C2×#12711yescharacteristic
333C23#53612yes—
334C23#53612yes—
335C23#57612yes—
336C23#57712yes—
337C23#57712yes—
338C23#57812yes—
339Q8—816no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—88no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#39014yes—
383C2×C4#39014yes—
384C2×C4C22×C22⋊C411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4#51912yes—
387C2×C4#51912yes—
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12411yescharacteristic
390C2×C4#53612yes—
391C2×C4#53612yes—
392C2×C4#53612yes—
393C2×C4#53612yes—
394C2×C4#53611yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#53611yescharacteristic
397C2×C4#53611yescharacteristic
398C2×C4#54112yes—
399C2×C4#54112yes—
400C2×C4#54112yes—
401C2×C4#54112yes—
402C8—28no—
403C8—42no—
404C8—42no—
405C22C2×#39011yescharacteristic
406C22#232712yes—
407C22#262112yes—
408C22#265811yescharacteristic
409C22#265811yescharacteristic
410C22#294612yes—
411C22#294712yes—
412C22#306212yes—
413C22#306312yes—
414C22#315411yescharacteristic
415C22#315411yescharacteristic
416C22#315511yescharacteristic
417C22#315611yescharacteristic
418C22#324114yes—
419C22#324214yes—
420C22#324314yes—
421C22#324414yes—
422C4—24no—
423C4—24no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—84no—
428C4—162no—
429C4—161no—
430C4—161no—
431C4#231012yes—
432C4#232711yescharacteristic
433C4#232711yescharacteristic
434C4#262111yescharacteristic
435C4#262111yescharacteristic
436C4#264212yes—
437C2#1427411yescharacteristic
438C2#2123411yescharacteristic
439C2#2123511yescharacteristic
440C2#2225514yes—
441C2#2988514yes—
442C2#3028711yescharacteristic
443C2#3028811yescharacteristic
444C2#3028911yescharacteristic
445C2#3029011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#501C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#19912
C2×#19902
C2×#18362
C2×#17462
#142832
#142822
#113792
#113782
#113772