Finite Group Databasebeta

← #193279

Subgroups — #193279

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7902C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2369—21no—
17#2369—21no—
18#2365C2212yes—
19#2365C2212yes—
20#2345—22no—
21Q8×C22⋊C4—22no—
22#2331—22no—
23#2316—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1990C2212yes—
31#1990C2212yes—
32#1988—22no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2212yes—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2212yes—
39#1854—22no—
40#1854—21no—
41#1854—21no—
42#1847—21no—
43#1836—21no—
44#1823—22no—
45#1776—21no—
46#1776—21no—
47#1556C418yescomplemented
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8Q814yes—
62C2×C4⋊Q8C2312yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#432C2×C414yes—
75#432C2×C414yes—
76#431—22no—
77#431—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—22no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8C2312yes—
94#391—24no—
95#391—24no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#382—21no—
112#380—24no—
113#380—21no—
114#380—21no—
115#380—21no—
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⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—24no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#340—24no—
137#340—24no—
138#340—22no—
139#340—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#332C2×C418yes—
145#332C2×C418yes—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—82no—
151C22×Q8—82no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8C4⋊C418yes—
168C4⋊Q8C22×C414yes—
169C4⋊Q8C2×Q814yes—
170C4×Q8—82no—
171C4×Q8—82no—
172C4×Q8—82no—
173C4×Q8—82no—
174#117—22no—
175#117—22no—
176#117—41no—
177#117—41no—
178#117—42no—
179#117—42no—
180#117—82no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
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—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C4⋊C418yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—41no—
221C2×C42C22×C412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#107—82no—
232#106—42no—
233#106—42no—
234#106—82no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241C4⋊C8C22×C414yes—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245#97—24no—
246#97—24no—
247C4×C8—22no—
248C4×C8—22no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—22no—
259#95—41no—
260#95—41no—
261C24#12011yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—28no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—88no—
272C2×Q8—88no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C4⋊C412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4C2×C4⋊C418yes—
347C4⋊C4#13712yes—
348C4⋊C4#13712yes—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—82no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42—84no—
363C42—84no—
364C42C2×C4⋊C412yes—
365C42C4×D412yes—
366C42C4×D412yes—
367C42C22⋊Q812yes—
368C42C22⋊Q812yes—
369C42#12312yes—
370C42C23×C411yescharacteristic
371C42C22×Q811yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23#38512yes—
381C23C2×#12011yescharacteristic
382Q8—44no—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—88no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—88no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#38512yes—
432C2×C4#38512yes—
433C2×C4#38512yes—
434C2×C4C22×C4⋊C411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57511yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57612yes—
445C2×C4#57712yes—
446C8—42no—
447C8—42no—
448C8—44no—
449C8—82no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22C2×#38511yescharacteristic
458C22#300411yescharacteristic
459C22#300511yescharacteristic
460C22#300811yescharacteristic
461C22#300911yescharacteristic
462C22#302011yescharacteristic
463C22#302011yescharacteristic
464C4—21no—
465C4—21no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—42no—
470C4—48no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4—81no—
475C4—82no—
476C4—161no—
477C4—161no—
478C4C4×#13711yescharacteristic
479C4C4×#13711yescharacteristic
480C4#294612yes—
481C4#307811yescharacteristic
482C4#307811yescharacteristic
483C4#308112yes—
484C2—42no—
485C2#1882811yescharacteristic
486C2#1882911yescharacteristic
487C2#1883211yescharacteristic
488C2#1883311yescharacteristic
489C2#2085711yescharacteristic
490C2#2087411yescharacteristic
491C2#2087611yescharacteristic
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. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
47. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219882
#218332
#213642
C2×#15562
#89262
#89252
#89232
#89222
#79022
#79002