Finite Group Databasebeta

← #381028

Subgroups — #381028

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1832C211yesmaximal, characteristic
3#13797C211yesmaximal, characteristic
4#11385C212yesmaximal
5#11384C212yesmaximal
6#11373C211yesmaximal, characteristic
7#11372C211yesmaximal, characteristic
8#11372C211yesmaximal, characteristic
9#11371C211yesmaximal, characteristic
10#10850C212yesmaximal
11#10849C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15#2660—24no—
16#2660C418yescomplemented
17C2×#398—22no—
18C2×#395—22no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×#376C2211yescharacteristic
25#1832—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—22no—
41#1434—22no—
42C23×Q8—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8C2×C414yes—
49C2×#125—22no—
50C2×#125—42no—
51C2×#125—42no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71#398—216no—
72#395—216no—
73#384—22no—
74#384—22no—
75#384C2×C414yes—
76#384C2×C414yes—
77#382—22no—
78#382—22no—
79#382C2×C414yes—
80#382C2×C414yes—
81C4×C4⋊C4—24no—
82C4×C4⋊C4Q818yescomplemented
83#376C2×C414yes—
84#376D414yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336—22no—
105C22×Q8—24no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—82no—
109C22×Q8—82no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—48no—
136C4⋊Q8—48no—
137#125—48no—
138#125—48no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C2×Q814yes—
181C2×C4⋊C4C2×Q814yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—28no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×Q814yes—
196C2×C42#4114yes—
197C4⋊C8—48no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95C22⋊C418yes—
206#95C22⋊C418yes—
207C24C22×D411yescharacteristic
208C2×Q8—48no—
209C2×Q8—88no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—88no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
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—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C414yes—
269C22×C4C22⋊Q814yes—
270C22×C4C22⋊Q814yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q812yes—
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)12yes—
276C22×C4H(5,𝔽2)12yes—
277C22×C42−1+412yes—
278C22×C42−1+412yes—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—88no—
313C4⋊C4—88no—
314C42—24no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—24no—
319C42—88no—
320C42C4×Q818yes—
321C42#12414yes—
322C42C4⋊Q814yes—
323C23C22×C22⋊C411yescharacteristic
324C23C2×C22⋊Q812yes—
325C23#53511yescharacteristic
326C23#53511yescharacteristic
327C23#53612yes—
328C23#57612yes—
329C23#57712yes—
330C23#57712yes—
331C23#57812yes—
332Q8—816no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#39118yes—
391C2×C4C2×C4×Q812yes—
392C2×C4#51812yes—
393C2×C4#51911yescharacteristic
394C2×C4#51911yescharacteristic
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×C4⋊Q811yescharacteristic
397C2×C4#53512yes—
398C2×C4#53612yes—
399C2×C4#54212yes—
400C2×C4#54212yes—
401C2×C4#54212yes—
402C2×C4#54212yes—
403C2×C4#54412yes—
404C2×C4#54412yes—
405C2×C4#54412yes—
406C2×C4#54412yes—
407C8—82no—
408C8—82no—
409C22C2×#39111yescharacteristic
410C22#230911yescharacteristic
411C22#231011yescharacteristic
412C22#262812yes—
413C22#262812yes—
414C22#289712yes—
415C22#289812yes—
416C22#308112yes—
417C22#308212yes—
418C22#308312yes—
419C22#308412yes—
420C22#322011yescharacteristic
421C22#322111yescharacteristic
422C22#322211yescharacteristic
423C22#322311yescharacteristic
424C22#322412yes—
425C22#322512yes—
426C22#324112yes—
427C22#324212yes—
428C22#324312yes—
429C22#324412yes—
430C4—28no—
431C4—42no—
432C4—42no—
433C4—44no—
434C4—84no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4#233112yes—
441C4#233112yes—
442C4#264012yes—
443C4#266012yes—
444C2#1428211yescharacteristic
445C2#2131512yes—
446C2#2181311yescharacteristic
447C2#2181411yescharacteristic
448C2#2184312yes—
449C2#2999012yes—
450C2#2999112yes—
451C2#2999212yes—
452C2#2999312yes—
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. #2660C4semidirect
82. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#18322
#137972
#113852
#113842
#113732
#113722
#113712
#108502
#108492