Finite Group Databasebeta

← #403662

Subgroups — #403662

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#2085C212yesmaximal
3C2×#2058C211yesmaximal, characteristic
4C2×#2057C211yesmaximal, characteristic
5#14276C211yesmaximal, characteristic
6#11208C211yesmaximal, characteristic
7#11208C211yesmaximal, characteristic
8#11207C212yesmaximal
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15#2331—28no—
16C2×#391—22no—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×#376C2211yescharacteristic
26#2085—28no—
27#2058—24no—
28#2057—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1441—22no—
40#1441—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—24no—
71#391—216no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389D418yes—
78#387—216no—
79#384—22no—
80#384—22no—
81#382—24no—
82#382—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85#376—24no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105#336—24no—
106#336Q818yescomplemented
107C22×Q8—24no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8—82no—
111C22×Q8—82no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2411yescharacteristic
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—
135C22×C8C2×Q814yes—
136C22×C8C2×Q814yes—
137C4⋊Q8—216no—
138C4⋊Q8—216no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
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⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4#4112yes—
182C2×C4⋊C4#4112yes—
183C2×C4⋊C4#4114yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—28no—
193C2×C42—82no—
194C2×C42—82no—
195C2×C42C2×D414yes—
196C2×C42C2×D412yes—
197C2×C42C2×Q814yes—
198C2×C42#4112yes—
199#107—48no—
200#106—48no—
201#95—28no—
202#95—24no—
203#95—24no—
204#95—28no—
205#95—24no—
206#95—24no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—82no—
211#95—82no—
212C24C22×D411yescharacteristic
213C2×Q8—216no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×Q8—88no—
217C2×Q8—88no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
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—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
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—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4#12014yes—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4C4⋊Q814yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×Q812yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C42−1+412yes—
284C22×C42−1+411yescharacteristic
285C22×C42−1+411yescharacteristic
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—216no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—88no—
324C4⋊C4#12118yes—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—88no—
332C23C2×#12011yescharacteristic
333C23C2×#12111yescharacteristic
334C23#52511yescharacteristic
335C23#52511yescharacteristic
336C23C2×C4⋊Q811yescharacteristic
337C23#53611yescharacteristic
338C23#53611yescharacteristic
339C23#57612yes—
340C23#57712yes—
341C23#57712yes—
342C23#57812yes—
343Q8—816no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4C2×#12112yes—
403C2×C4#52512yes—
404C2×C4#53312yes—
405C2×C4#53312yes—
406C2×C4#54112yes—
407C2×C4#54112yes—
408C2×C4#54712yes—
409C2×C4#54712yes—
410C2×C4#54811yescharacteristic
411C2×C4#54811yescharacteristic
412C2×C4D4×Q812yes—
413C2×C4D4×Q812yes—
414C2×C4D4×Q812yes—
415C2×C4#55211yescharacteristic
416C2×C4#55211yescharacteristic
417C2×C4#55712yes—
418C8—44no—
419C8—44no—
420C22#239911yescharacteristic
421C22#240611yescharacteristic
422C22#243411yescharacteristic
423C22#253611yescharacteristic
424C22#254611yescharacteristic
425C22#263411yescharacteristic
426C22#284711yescharacteristic
427C22#301212yes—
428C22#301312yes—
429C22#304111yescharacteristic
430C22#304211yescharacteristic
431C22#304311yescharacteristic
432C22#304411yescharacteristic
433C22#311812yes—
434C22#311812yes—
435C22#312112yes—
436C22#312112yes—
437C22#316312yes—
438C22#316412yes—
439C22#324112yes—
440C22#324212yes—
441C22#324312yes—
442C22#324412yes—
443C4—44no—
444C4—42no—
445C4—48no—
446C4—42no—
447C4—44no—
448C4—84no—
449C4—82no—
450C4—82no—
451C4—161no—
452C4—161no—
453C4#242912yes—
454C4#242912yes—
455C4#246212yes—
456C4#246212yes—
457C2#1488011yescharacteristic
458C2#2384212yes—
459C2#2407512yes—
460C2#2582212yes—
461C2#2821011yescharacteristic
462C2#2821311yescharacteristic
463C2#2869611yescharacteristic
464C2#2869711yescharacteristic
465C2#3010012yes—
466C2#3291711yescharacteristic
467C2#3291811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
106. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20852
C2×#20582
C2×#20572
#142762
#112082
#112072
#109082
#109072
#107962