Finite Group Databasebeta

← #193264

Subgroups — #193264

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8928C212yesmaximal, complemented
8#8927C212yesmaximal, complemented
9#8927C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#419—22no—
16C2×C4×M16C2211yescharacteristic
17#2369C2212yes—
18#2369C2212yes—
19#2365—21no—
20#2365—21no—
21#2345—22no—
22Q8×C22⋊C4—22no—
23#2331—22no—
24#2316—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1960—22no—
32#1956C2212yes—
33#1956C2212yes—
34#1948—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2212yes—
38#1947—21no—
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2212yes—
42#1854—21no—
43#1854—21no—
44#1809—22no—
45C4×#107—21no—
46C4×#106—21no—
47#1548C418yescomplemented
48C2×#333C2212yes—
49C2×#333C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8D412yes—
55C22×M16—21no—
56C22×M16—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8Q814yes—
64C2×C4⋊Q8C2312yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431C2×C414yes—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—22no—
82#423—22no—
83#423C2×C414yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88#419—48no—
89C2×#103—24no—
90C2×#103—24no—
91C2×C22⋊C8—21no—
92C4×M16—22no—
93C4×M16—22no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#391—22no—
98#391—22no—
99#391—24no—
100#391—24no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#387—24no—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#382—21no—
115#380—24no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380D412yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#340—22no—
140#340—22no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#333C2×C418yes—
146#333C2×C418yes—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C4⋊C418yes—
152C22×Q8C2×D414yes—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—81no—
162C2×M16—81no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C4⋊C418yes—
175C4⋊Q8C22×C414yes—
176C4⋊Q8C2×Q814yes—
177C4×Q8—84no—
178C4×Q8—84no—
179#117—22no—
180#117—22no—
181#117—41no—
182#117—41no—
183#117—42no—
184#117—42no—
185#117—44no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—41no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234#107—42no—
235#107—42no—
236#107—82no—
237#106—42no—
238#106—42no—
239#106—82no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C22×C414yes—
245C4⋊C8C22×C412yes—
246C4⋊C8C22×C412yes—
247#103—48no—
248#103—48no—
249C22⋊C8—24no—
250#97—42no—
251C4×C8—24no—
252C4×C8—22no—
253C4×C8—22no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—41no—
265#95—41no—
266C24#12011yescharacteristic
267C2×Q8—24no—
268C2×Q8—28no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—416no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—88no—
275C2×Q8C2×C4⋊C418yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C2×C4⋊C412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12312yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310M16—48no—
311M16—84no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4#13712yes—
357C4⋊C4#13712yes—
358C22⋊C4—24no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—82no—
362C42—24no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42—84no—
373C42—84no—
374C42—82no—
375C42C2×C4⋊C412yes—
376C42C4×D412yes—
377C42C4×D412yes—
378C42C22⋊Q812yes—
379C42C22⋊Q812yes—
380C42#12312yes—
381C42C23×C411yescharacteristic
382C42C22×Q811yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23#38512yes—
392C23C2×#12011yescharacteristic
393Q8—48no—
394Q8—416no—
395Q8—84no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—28no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4#45312yes—
444C2×C4#45512yes—
445C2×C4C22×C4⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57511yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57811yescharacteristic
458C2×C4#57811yescharacteristic
459C8—44no—
460C8—42no—
461C8—42no—
462C8—44no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22C2×#38511yescharacteristic
471C22#300411yescharacteristic
472C22#300511yescharacteristic
473C22#301311yescharacteristic
474C22#301311yescharacteristic
475C22C2×#45311yescharacteristic
476C22C2×#45511yescharacteristic
477C4—21no—
478C4—21no—
479C4—22no—
480C4—22no—
481C4—48no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4—81no—
487C4—82no—
488C4—161no—
489C4—161no—
490C4#185614yes—
491C4C4×#13711yescharacteristic
492C4C4×#13711yescharacteristic
493C4#294911yescharacteristic
494C4#295011yescharacteristic
495C4#307811yescharacteristic
496C4#307811yescharacteristic
497C4#308211yescharacteristic
498C4#308211yescharacteristic
499C2—42no—
500C2#1856711yescharacteristic
501C2#1856911yescharacteristic
502C2#1857511yescharacteristic
503C2#1857711yescharacteristic
504C2C2×#185611yescharacteristic
505C2#2087411yescharacteristic
506C2#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. #8929C2semidirect
7. #8928C2semidirect
8. #8927C2semidirect
9. #8927C2semidirect
47. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219642
#213622
#212022
C2×#15482
#89292
#89282
#89272
#79032
#78962