Finite Group Databasebeta

← #108283

Subgroups — #108283

Order 512.

857 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34235C211yesmaximal, characteristic
2#17749C211yesmaximal, characteristic
3#17545C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2365—21no—
13#2365—21no—
14#2360—22no—
15#2356—21no—
16#2351—21no—
17#2343—22no—
18#2336—22no—
19#2320C412yescomplemented
20C4×C22⋊Q8—22no—
21#2283C2211yescharacteristic
22#2276C412yescomplemented
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#339—22no—
26#1559C414yescomplemented
27#1550—22no—
28#1548—21no—
29#1546—21no—
30#1508—22no—
31#1506—22no—
32#1503C414yescomplemented
33#1286—21no—
34#1286—21no—
35#1285—22no—
36#1284—22no—
37C22×M16—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44#533D412yes—
45#533Q812yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8D412yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76#407—21no—
77#407—21no—
78C4×M16—21no—
79C4×M16—21no—
80#391—22no—
81#391—22no—
82#389—24no—
83#389—22no—
84#389—22no—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#385—24no—
90#383—21no—
91#383—22no—
92#382—22no—
93#382D411yescharacteristic
94#382D411yescharacteristic
95#381D412yes—
96#380—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#376—22no—
114#376—21no—
115#376C2×C412yes—
116C2×#95—21no—
117C43—21no—
118#339—44no—
119#333—24no—
120#332—24no—
121#328—28no—
122C22×Q8—21no—
123C22×Q8C22⋊C412yes—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—22no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—42no—
133C2×M16—42no—
134C22×C8—41no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C22⋊C412yes—
140C4⋊Q8C22⋊C412yes—
141C4⋊Q8C4⋊C412yes—
142C4⋊Q8C4⋊C412yes—
143#125—22no—
144#125—22no—
145#125—22no—
146#125—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8C22⋊C414yes—
154C22⋊Q8C4⋊C414yes—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—48no—
162C4×Q8—82no—
163C4×Q8—82no—
164C4×Q8C4214yes—
165C4×Q8C4⋊C414yes—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—41no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—41no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—41no—
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q811yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228#97—22no—
229#97—41no—
230C4×C8—22no—
231C4×C8—41no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—41no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24#12011yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8#9514yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291M16—44no—
292M16—82no—
293M16—82no—
294M16—81no—
295M16—81no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—82no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—84no—
322C4⋊C4#9514yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C42—24no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—44no—
333C42—48no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—82no—
339C42—82no—
340C42—84no—
341C42C2×C4211yescharacteristic
342C42C2×C22⋊C411yescharacteristic
343C42C2×C4⋊C411yescharacteristic
344C42C2×C4⋊C411yescharacteristic
345C42#11711yescharacteristic
346C42#11711yescharacteristic
347C42C4×D411yescharacteristic
348C42C4×D411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42#12111yescharacteristic
352C42#12111yescharacteristic
353C42C22⋊Q811yescharacteristic
354C42C22⋊Q811yescharacteristic
355C42#12311yescharacteristic
356C42#12311yescharacteristic
357C23—21no—
358C23—21no—
359C23—22no—
360C23—41no—
361C23#37911yescharacteristic
362C23#38511yescharacteristic
363C23#38611yescharacteristic
364Q8—44no—
365Q8—44no—
366Q8—42no—
367Q8—42no—
368Q8—42no—
369Q8—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—88no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4C2×#9511yescharacteristic
414C2×C4C4×C22⋊C411yescharacteristic
415C2×C4C4×C22⋊C411yescharacteristic
416C2×C4#38011yescharacteristic
417C2×C4#38011yescharacteristic
418C2×C4#38111yescharacteristic
419C2×C4#38511yescharacteristic
420C2×C4#38511yescharacteristic
421C2×C4#38511yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#45311yescharacteristic
426C2×C4#45411yescharacteristic
427C2×C4#45511yescharacteristic
428C2×C4#45611yescharacteristic
429C8—82no—
430C8—82no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22C22⋊#9511yescharacteristic
442C22#179211yescharacteristic
443C22#179311yescharacteristic
444C22#190512yes—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—44no—
456C4—44no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—84no—
461C4#182712yes—
462C4#185911yescharacteristic
463C4#186011yescharacteristic
464C4#188211yescharacteristic
465C4#188311yescharacteristic
466C2—21no—
467C2—21no—
468C2—22no—
469C2—41no—
470C2#1075811yescharacteristic
471C2#1078111yescharacteristic
472C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
19. #2320C4semidirect
22. #2276C4semidirect
26. #1559C4semidirect
32. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#342352
#177492
#175452
#108522
#79602
#79502
#79402