Finite Group Databasebeta

← #108282

Subgroups — #108282

Order 512.

985 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34236C211yesmaximal, characteristic
2#17745C211yesmaximal, characteristic
3#17544C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic, complemented
5#7960C211yesmaximal, characteristic, complemented
6#7950C211yesmaximal, characteristic, complemented
7#7940C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2369—21no—
13#2366—21no—
14#2361—22no—
15#2357—21no—
16#2349—21no—
17#2339—22no—
18#2337—22no—
19#2319C412yescomplemented
20C4×#121—22no—
21#2283C2211yescharacteristic
22#2275C412yescomplemented
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#339—22no—
26#1557C414yescomplemented
27#1549—21no—
28#1547—22no—
29#1545—21no—
30#1507—22no—
31#1505—22no—
32#1502C414yescomplemented
33#1286—21no—
34#1286—21no—
35#1285—22no—
36#1284—22no—
37C22×#41—21no—
38C22×M16—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532C2×C412yes—
44#532C2×C412yes—
45#532D412yes—
46#532Q812yes—
47C2×C4⋊Q8D411yescharacteristic
48C2×#127D411yescharacteristic
49C2×#124—21no—
50C2×#121—21no—
51C2×#121D412yes—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4C2×C412yes—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yescomplemented
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yescomplemented
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423C2×C412yescomplemented
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77#407—21no—
78#407—21no—
79C4×M16—21no—
80C4×M16—21no—
81#391—24no—
82#390—22no—
83#390—22no—
84#389—22no—
85#389—22no—
86#388—24no—
87#388—24no—
88#386—24no—
89#386—24no—
90#385—24no—
91#383—21no—
92#382D411yescharacteristic
93#382D411yescharacteristic
94#381—22no—
95#381D412yes—
96#380—21no—
97#380—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4C2×C412yescomplemented
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113#376—21no—
114#376C2×C412yes—
115C2×#95—21no—
116C2×#95—22no—
117C43—21no—
118#339—44no—
119#332—24no—
120#331—24no—
121#327—28no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—84no—
127C22×Q8—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C22⋊C412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—22no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—41no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C22⋊C412yes—
148C4⋊Q8C4⋊C412yes—
149#127—21no—
150#127—21no—
151#127C22⋊C412yes—
152#127C4⋊C412yes—
153#124—22no—
154#124—22no—
155#124—22no—
156#124—22no—
157#121—22no—
158#121—22no—
159#121—22no—
160#121—22no—
161#121—22no—
162#121—22no—
163#121C22⋊C414yes—
164#121C4⋊C414yes—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—48no—
172C4×D4—82no—
173C4×D4—82no—
174C4×D4—88no—
175C4×D4C4214yes—
176C4×D4C4⋊C414yes—
177#117—22no—
178#117—22no—
179#117—41no—
180#117—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—41no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—41no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—41no—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q811yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232#97—22no—
233#97—41no—
234C4×C8—22no—
235C4×C8—41no—
236#95—24no—
237#95—24no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—41no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24—22no—
250C24#12011yescharacteristic
251#41—84no—
252#41—84no—
253#41—84no—
254#41—84no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—82no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—44no—
266C2×D4—48no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—82no—
272C2×D4#9514yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—82no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4C22⋊Q811yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312M16—44no—
313M16—82no—
314M16—82no—
315M16—81no—
316M16—81no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—82no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—84no—
334C4⋊C4#9514yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—48no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C42—42no—
346C42—42no—
347C42—44no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—82no—
353C42—82no—
354C42—84no—
355C42C2×C4211yescharacteristic
356C42C2×C22⋊C411yescharacteristic
357C42C2×C4⋊C411yescharacteristic
358C42C2×C4⋊C411yescharacteristic
359C42#11711yescharacteristic
360C42#11711yescharacteristic
361C42C4×D411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42#12111yescharacteristic
366C42#12111yescharacteristic
367C42C22⋊Q811yescharacteristic
368C42C22⋊Q811yescharacteristic
369C42#12311yescharacteristic
370C42#12311yescharacteristic
371C23—21no—
372C23—21no—
373C23—22no—
374C23—24no—
375C23—41no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—42no—
380C23#37911yescharacteristic
381C23#38511yescharacteristic
382C23#38611yescharacteristic
383Q8—82no—
384Q8—82no—
385D4—44no—
386D4—48no—
387D4—48no—
388D4—44no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—82no—
394D4—82no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—88no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—168no—
438C2×C4C2×#9511yescharacteristic
439C2×C4C4×C22⋊C411yescharacteristic
440C2×C4C4×C22⋊C411yescharacteristic
441C2×C4#38011yescharacteristic
442C2×C4#38011yescharacteristic
443C2×C4#38111yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38511yescharacteristic
446C2×C4#38511yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#45311yescharacteristic
451C2×C4#45411yescharacteristic
452C2×C4#45511yescharacteristic
453C2×C4#45611yescharacteristic
454C8—82no—
455C8—82no—
456C8—82no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—84no—
470C22—84no—
471C22C22⋊#9511yescharacteristic
472C22#179211yescharacteristic
473C22#179311yescharacteristic
474C22#190412yes—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—44no—
485C4—44no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4#182712yes—
491C4#185911yescharacteristic
492C4#186011yescharacteristic
493C4#188211yescharacteristic
494C4#188311yescharacteristic
495C2—21no—
496C2—21no—
497C2—22no—
498C2—41no—
499C2—84no—
500C2#1075811yescharacteristic
501C2#1078111yescharacteristic
502C2#1078511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10852C2semidirect
5. #7960C2semidirect
6. #7950C2semidirect
7. #7940C2semidirect
19. #2319C4semidirect
22. #2275C4semidirect
26. #1557C4semidirect
32. #1502C4semidirect
64. #432C2×C4semidirect
68. #431C2×C4semidirect
74. #423C2×C4semidirect
106. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342362
#177452
#175442
#108522
#79602
#79502
#79402