Finite Group Databasebeta

← #381552

Subgroups — #381552

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic
3C2×#1928C212yesmaximal
4#14261C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11324C211yesmaximal, characteristic
8#11067C212yesmaximal
9#11050C211yesmaximal, characteristic
10#10806C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2330—28no—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389—22no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#2026C418yescomplemented
28#1928—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125—22no—
48C2×#125C2×C414yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—24no—
63C2×#107—42no—
64C2×#107—42no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103C2×C418yes—
71#391—22no—
72#391—24no—
73#391—42no—
74#391C2×C414yes—
75#389—216no—
76#389—24no—
77#389—24no—
78#389—42no—
79#387—216no—
80#384—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4D418yes—
83#376—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#340C2×C418yes—
100#336—24no—
101#336—22no—
102#336—42no—
103#336C2×C414yes—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8C22×C412yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
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—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C22×C414yes—
139#125—216no—
140#125—216no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
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—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C414yes—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4#4114yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—28no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42C22×C412yes—
189C2×C42C2×D414yes—
190C2×C42#4112yes—
191#107—48no—
192#103—216no—
193#103—48no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—24no—
198#95—24no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—82no—
204#95—82no—
205C24C22×D411yescharacteristic
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—24no—
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—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C4×D414yes—
281C22×C4#12014yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)12yes—
288C22×C42−1+411yescharacteristic
289C22×C42−1+411yescharacteristic
290C22×C42−1+412yes—
291C22×C42−1+411yescharacteristic
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—232no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C4⋊C4—84no—
323C4⋊C4C4×D418yes—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—42no—
329C42—42no—
330C42—88no—
331C23C2×C4×D411yescharacteristic
332C23C2×#12011yescharacteristic
333C23#52511yescharacteristic
334C23#53511yescharacteristic
335C23#53611yescharacteristic
336C23#53611yescharacteristic
337C23#53711yescharacteristic
338C23#57612yes—
339C23#57712yes—
340C23#57812yes—
341C23#57812yes—
342Q8—44no—
343Q8—44no—
344Q8—82no—
345Q8—82no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4C2×C4×D412yes—
407C2×C4#51911yescharacteristic
408C2×C4#52012yes—
409C2×C4#52011yescharacteristic
410C2×C4#52512yes—
411C2×C4#53512yes—
412C2×C4#53612yes—
413C2×C4#53612yes—
414C2×C4#53712yes—
415C2×C4#54112yes—
416C2×C4#54111yescharacteristic
417C2×C4#54212yes—
418C2×C4#54212yes—
419C2×C4#54212yes—
420C2×C4#54211yescharacteristic
421C2×C4#54212yes—
422C8—44no—
423C8—82no—
424C22C4×#12011yescharacteristic
425C22#232311yescharacteristic
426C22#232611yescharacteristic
427C22#261411yescharacteristic
428C22#261511yescharacteristic
429C22#263211yescharacteristic
430C22#263711yescharacteristic
431C22#294712yes—
432C22#295112yes—
433C22#301112yes—
434C22#301212yes—
435C22#312012yes—
436C22#312412yes—
437C22#322011yescharacteristic
438C22#322111yescharacteristic
439C22#322311yescharacteristic
440C22#322311yescharacteristic
441C22#324212yes—
442C22#324212yes—
443C22#324312yes—
444C22#324312yes—
445C22#325712yes—
446C22#326112yes—
447C4—44no—
448C4—44no—
449C4—48no—
450C4—82no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—84no—
455C4—161no—
456C4—161no—
457C4#232312yes—
458C4#232612yes—
459C4#263212yes—
460C4#263712yes—
461C2#1430111yescharacteristic
462C2#2146512yes—
463C2#2212412yes—
464C2#2222811yescharacteristic
465C2#2223111yescharacteristic
466C2#2964912yes—
467C2#2967411yescharacteristic
468C2#2967511yescharacteristic
469C2#3007211yescharacteristic
470C2#3007311yescharacteristic
471C2#3016412yes—
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
27. #2026C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#20262
C2×#19282
#142612
#113852
#113282
#113242
#110672
#110502
#108062
#107962