Finite Group Databasebeta

← #192679

Subgroups — #192679

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#20760C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8892C212yesmaximal, complemented
9#8891C212yesmaximal, complemented
10#8041C211yesmaximal, characteristic
11#7897C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2382—22no—
17#2365—21no—
18#2365—21no—
19#2355—21no—
20#2355C2212yes—
21Q8×C4⋊C4—21no—
22Q8×C4⋊C4C2212yes—
23#2323—22no—
24#2323—22no—
25#2314—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1988—21no—
30#1981—21no—
31#1948—21no—
32#1948C2212yes—
33#1947—21no—
34#1947C2212yes—
35#1894—22no—
36#1835—22no—
37#1834C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1809—22no—
42#1803C2212yes—
43#1803C2212yes—
44#1735—22no—
45#1493C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×M16—21no—
53C22×M16—21no—
54C23×C8—21no—
55#533—22no—
56#533—22no—
57#533C2×C414yes—
58#533Q814yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8D412yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D412yes—
72C22×C42C2311yescharacteristic
73C2×#107—22no—
74C2×#106—22no—
75#423—22no—
76#423—22no—
77#423—22no—
78#423C2×C414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×#103—22no—
84C2×#103—22no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8C2×C414yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#391—22no—
92#389—22no—
93#389—24no—
94#389—22no—
95#389—24no—
96#388—24no—
97#388—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#384—22no—
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—22no—
110#382—21no—
111#382—21no—
112#382D412yes—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380D412yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2312yes—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137Q8⋊C8C2×C418yes—
138Q8⋊C8C2×C418yes—
139C22×Q8—21no—
140C22×Q8C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—42no—
146C2×M16—81no—
147C2×M16—81no—
148C2×M16—81no—
149C2×M16—81no—
150C22×C8—24no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8C4⋊C414yes—
160C4⋊Q8C4⋊C414yes—
161#125—24no—
162#125—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8C4⋊C418yes—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—82no—
172C4×Q8—82no—
173C4×Q8—84no—
174C4×Q8C22×C414yes—
175C4×Q8C2×Q814yes—
176#117—24no—
177#117—24no—
178#117—44no—
179#117—44no—
180#117—81no—
181#117—81no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—41no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—41no—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×Q812yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—82no—
227#106—82no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233C4⋊C8C22×C414yes—
234#103—82no—
235#103—82no—
236C4×C8—24no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8C22×C412yes—
240C4×C8C22×C412yes—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251C24#12011yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8C2×C4⋊C414yes—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×C4⋊C412yes—
280C22×C4C4×D412yes—
281C22×C4C4×D412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4#12312yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—84no—
292M16—84no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—48no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4C2×C4⋊C414yes—
344C22⋊C4—28no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—24no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—44no—
353C42—84no—
354C42—84no—
355C42—82no—
356C42C2×C4⋊C412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42C22⋊Q812yes—
360C42C22⋊Q812yes—
361C42#12312yes—
362C42C23×C411yescharacteristic
363C42C22×Q811yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23#38512yes—
373C23C2×#12011yescharacteristic
374Q8—24no—
375Q8—24no—
376Q8—42no—
377Q8—42no—
378Q8—48no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4#38512yes—
420C2×C4#38512yes—
421C2×C4#38512yes—
422C2×C4#45412yes—
423C2×C4#45512yes—
424C2×C4C22×C4⋊C411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4#57611yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57811yescharacteristic
436C2×C4#57811yescharacteristic
437C2×C4#57811yescharacteristic
438C8—42no—
439C8—42no—
440C8—44no—
441C8—82no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22C2×#38511yescharacteristic
450C22#301211yescharacteristic
451C22#301311yescharacteristic
452C22#301311yescharacteristic
453C22#301311yescharacteristic
454C22C2×#45511yescharacteristic
455C22C2×#45411yescharacteristic
456C4—22no—
457C4—21no—
458C4—21no—
459C4—22no—
460C4—44no—
461C4—44no—
462C4—82no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—161no—
467C4—161no—
468C4#186014yes—
469C4#291211yescharacteristic
470C4#291311yescharacteristic
471C4#294911yescharacteristic
472C4#295011yescharacteristic
473C4#308211yescharacteristic
474C4#308211yescharacteristic
475C4#309611yescharacteristic
476C4#309711yescharacteristic
477C2—42no—
478C2#1869211yescharacteristic
479C2#1869311yescharacteristic
480C2#1869711yescharacteristic
481C2#1869811yescharacteristic
482C2#2084911yescharacteristic
483C2C2×#186011yescharacteristic
484C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
45. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219642
#218942
#207602
C2×#14932
#88962
#88952
#88922
#88912
#80412
#78972