Finite Group Databasebeta

← #192771

Subgroups — #192771

Order 512.

956 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
1#34187C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21964C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8895C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8887C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2356—22no—
17#2356—22no—
18#2355—21no—
19#2355—21no—
20#2355C2212yes—
21#2355C2212yes—
22#2313—22no—
23#2310—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2283—22no—
28#2274—21no—
29#1949—22no—
30#1948—21no—
31#1948C2212yes—
32#1947—21no—
33#1947C2212yes—
34#1854—22no—
35#1845C2211yescharacteristic
36#1844C2211yescharacteristic
37#1834C2211yescharacteristic
38#1834C2211yescharacteristic
39#1823—21no—
40#1809—22no—
41#1803C2212yes—
42#1803C2212yes—
43#1787—21no—
44#1763—22no—
45#1487C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×Q8D411yescharacteristic
53C22×M16—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C414yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8Q814yes—
61C2×C4×Q8C2312yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#432—22no—
70#432—22no—
71#431—22no—
72#431C2×C414yes—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—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×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×#97—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#383—24no—
105#382—22no—
106#382—22no—
107#382—21no—
108#382—21no—
109#382—21no—
110#381—24no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
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⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130#376—24no—
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×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8C4⋊C414yes—
144C22×Q8C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—81no—
152C2×M16—81no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C4⋊C418yes—
165C4×Q8C22×C414yes—
166C4×Q8C2×Q814yes—
167#117—44no—
168#117—44no—
169#117—82no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C4⋊C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—41no—
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q812yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216#107—82no—
217#107—82no—
218#106—82no—
219#106—82no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8C22×C414yes—
226#103—44no—
227#103—44no—
228C22⋊C8—24no—
229C22⋊C8—42no—
230#97—42no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8C22×C412yes—
234C4×C8C22×C412yes—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—48no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8C2×C4⋊C414yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C2×C4⋊C412yes—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4#12312yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—84no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
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—28no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4C2×C4⋊C414yes—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—82no—
339C42—24no—
340C42—24no—
341C42—28no—
342C42—44no—
343C42—44no—
344C42—82no—
345C42—84no—
346C42—84no—
347C42C2×C4⋊C412yes—
348C42C4×D412yes—
349C42C4×D412yes—
350C42C22⋊Q812yes—
351C42C22⋊Q812yes—
352C42#12312yes—
353C42C23×C411yescharacteristic
354C42C22×Q811yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23#38512yes—
364C23C2×#12011yescharacteristic
365Q8—24no—
366Q8—24no—
367Q8—48no—
368Q8—42no—
369Q8—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
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—84no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4#38512yes—
413C2×C4#38512yes—
414C2×C4#38512yes—
415C2×C4C22×C4⋊C411yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C22⋊Q811yescharacteristic
420C2×C4C2×#12311yescharacteristic
421C2×C4#57512yes—
422C2×C4#57512yes—
423C2×C4#57711yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57811yescharacteristic
426C2×C4#57811yescharacteristic
427C8—42no—
428C8—42no—
429C8—44no—
430C8—82no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22C2×#38511yescharacteristic
439C22#300511yescharacteristic
440C22#300511yescharacteristic
441C22#301311yescharacteristic
442C22#301311yescharacteristic
443C22#302111yescharacteristic
444C22#302211yescharacteristic
445C4—22no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—44no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4#291112yes—
459C4#294911yescharacteristic
460C4#295011yescharacteristic
461C4#308211yescharacteristic
462C4#308211yescharacteristic
463C4#309312yes—
464C2—42no—
465C2#1884011yescharacteristic
466C2#1884111yescharacteristic
467C2#1884811yescharacteristic
468C2#1884911yescharacteristic
469C2#2084511yescharacteristic
470C2#2086511yescharacteristic
471C2#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. #8895C2semidirect
7. #8891C2semidirect
8. #8887C2semidirect
9. #8883C2semidirect
45. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#219842
#219642
#207402
C2×#14872
#88952
#88912
#88872
#88832
#80422
#79002