Finite Group Databasebeta

← #193278

Subgroups — #193278

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8928C212yesmaximal, complemented
8#8927C212yesmaximal, complemented
9#8927C212yesmaximal, complemented
10#7902C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2369C2212yes—
17#2369C2212yes—
18#2365—21no—
19#2365—21no—
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#1981—22no—
31#1956C2212yes—
32#1956C2212yes—
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#1787—22no—
45#1776—21no—
46#1776—21no—
47#1558C418yescomplemented
48C2×#333C2212yes—
49C2×#333C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8D412yes—
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⋊C4—21no—
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×#103—24no—
89C2×#103—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8C2312yes—
96#391—22no—
97#391—22no—
98#391—24no—
99#391—24no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#387—24no—
107#387—24no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#382—21no—
114#380—24no—
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—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—24no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—21no—
138#340—22no—
139#340—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#333C2×C418yes—
145#333C2×C418yes—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—82no—
149C22×Q8—82no—
150C22×Q8C4⋊C418yes—
151C22×Q8C2×D414yes—
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—84no—
171C4×Q8—84no—
172#117—22no—
173#117—22no—
174#117—41no—
175#117—41no—
176#117—42no—
177#117—42no—
178#117—82no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
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—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—41no—
219C2×C42C22×C412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42C2411yescharacteristic
227#107—42no—
228#107—42no—
229#107—82no—
230#106—42no—
231#106—42no—
232#106—82no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C414yes—
240#103—84no—
241#103—84no—
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—28no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—88no—
269C2×Q8—88no—
270C2×Q8C2×C4⋊C418yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C4⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4C22⋊Q812yes—
298C22×C4C22⋊Q812yes—
299C22×C4#12312yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4#13712yes—
345C4⋊C4#13712yes—
346C22⋊C4—24no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—82no—
350C42—24no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—82no—
359C42—84no—
360C42—84no—
361C42C2×C4⋊C412yes—
362C42C4×D412yes—
363C42C4×D412yes—
364C42C22⋊Q812yes—
365C42C22⋊Q812yes—
366C42#12312yes—
367C42C23×C411yescharacteristic
368C42C22×Q811yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23#38512yes—
378C23C2×#12011yescharacteristic
379Q8—48no—
380Q8—416no—
381Q8—84no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—88no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4#38512yes—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C2×C4#57511yescharacteristic
438C2×C4#57511yescharacteristic
439C2×C4#57712yes—
440C2×C4#57812yes—
441C8—42no—
442C8—42no—
443C8—44no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22C2×#38511yescharacteristic
453C22#300411yescharacteristic
454C22#300511yescharacteristic
455C22#301011yescharacteristic
456C22#301111yescharacteristic
457C22#301911yescharacteristic
458C22#301911yescharacteristic
459C4—21no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—48no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4—161no—
472C4—161no—
473C4C4×#13711yescharacteristic
474C4C4×#13711yescharacteristic
475C4#294712yes—
476C4#307811yescharacteristic
477C4#307811yescharacteristic
478C4#308212yes—
479C2—42no—
480C2#1881611yescharacteristic
481C2#1881711yescharacteristic
482C2#1882011yescharacteristic
483C2#1882111yescharacteristic
484C2#2085611yescharacteristic
485C2#2087411yescharacteristic
486C2#2087711yescharacteristic
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. #1558C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219882
#218042
#213642
C2×#15582
#89292
#89282
#89272
#79022
#79002