Finite Group Databasebeta

← #116817

Subgroups — #116817

Order 512.

635 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#21169C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6C2×#1301C211yesmaximal, characteristic
7#7924C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#427—21no—
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1995—22no—
17#1994—22no—
18#1949—22no—
19#1948—21no—
20#1947—21no—
21#1735—21no—
22#1733—21no—
23#1733—21no—
24#1520—22no—
25#1518—22no—
26#1503—22no—
27#1481—21no—
28#1481—21no—
29#1481C414yescomplemented
30C2×Q8⋊C8—22no—
31#1303—22no—
32#1301—22no—
33#1282—22no—
34C22×M16—21no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533D412yes—
44C2×C4⋊Q8Q811yescharacteristic
45C2×C4⋊Q8Q811yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117D411yescharacteristic
54C2×#117D411yescharacteristic
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#427—42no—
65#427—42no—
66C2×#107—22no—
67C2×#106—22no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423D412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C22⋊C8—21no—
84C2×C22⋊C8D412yes—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#384—21no—
88#382D412yes—
89#380—21no—
90#380—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97#376—21no—
98C43—21no—
99#340—22no—
100#340—22no—
101#339—24no—
102#339—24no—
103#328—28no—
104Q8⋊C8—24no—
105Q8⋊C8—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8C4⋊C412yes—
135C4⋊Q8C4⋊C412yes—
136#125—22no—
137#125—22no—
138#125—41no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—22no—
161#117—22no—
162#117—21no—
163#117—21no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117SD1612yes—
171#117Q1612yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q811yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205#107—44no—
206#106—44no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8C22⋊C412yes—
220C4⋊C8C22⋊C412yes—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C4×C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229#95—22no—
230#95—22no—
231#95—41no—
232#95—41no—
233C24#12111yescharacteristic
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—82no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4#11711yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12711yescharacteristic
265C22×C4C2×SD1611yescharacteristic
266C22×C4C2×Q1611yescharacteristic
267C22×C4#13511yescharacteristic
268C22×C4#13511yescharacteristic
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4#13711yescharacteristic
271C22×C4#13711yescharacteristic
272C22×C4#13711yescharacteristic
273M16—42no—
274M16—42no—
275M16—81no—
276M16—81no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—82no—
291C2×C8—81no—
292C2×C8—81no—
293C2×C8—82no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—22no—
323C42—22no—
324C42—42no—
325C42—42no—
326C42—41no—
327C42—41no—
328C42—82no—
329C42—82no—
330C42—81no—
331C42—81no—
332C42C2×C22⋊C411yescharacteristic
333C42C2×C4⋊C411yescharacteristic
334C42C4×D411yescharacteristic
335C42C4×D411yescharacteristic
336C42#12111yescharacteristic
337C42#12111yescharacteristic
338C42#12111yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42#12311yescharacteristic
343C42#12311yescharacteristic
344C42#12311yescharacteristic
345C42#12311yescharacteristic
346C42#12411yescharacteristic
347C42#12411yescharacteristic
348C23—41no—
349C23—41no—
350C23#38811yescharacteristic
351C23#39211yescharacteristic
352C23#39611yescharacteristic
353C23#46511yescharacteristic
354C23#46711yescharacteristic
355C23#46811yescharacteristic
356C23#47011yescharacteristic
357Q8—44no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#38011yescharacteristic
387C2×C4#38511yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38611yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#39011yescharacteristic
392C2×C4#39311yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39411yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4#39911yescharacteristic
398C2×C4C4×SD1611yescharacteristic
399C2×C4C4×Q1611yescharacteristic
400C2×C4#44011yescharacteristic
401C2×C4#44111yescharacteristic
402C2×C4#44811yescharacteristic
403C2×C4#44911yescharacteristic
404C2×C4#45011yescharacteristic
405C2×C4#45111yescharacteristic
406C2×C4#45111yescharacteristic
407C2×C4#45211yescharacteristic
408C2×C4#45211yescharacteristic
409C2×C4#45211yescharacteristic
410C2×C4#46111yescharacteristic
411C2×C4#46211yescharacteristic
412C2×C4#46311yescharacteristic
413C2×C4#46411yescharacteristic
414C2×C4#47511yescharacteristic
415C2×C4#47711yescharacteristic
416C2×C4#47911yescharacteristic
417C2×C4#47911yescharacteristic
418C2×C4#48711yescharacteristic
419C2×C4#48811yescharacteristic
420C2×C4#48911yescharacteristic
421C2×C4#49111yescharacteristic
422C8—42no—
423C8—42no—
424C8—82no—
425C8—82no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22#143911yescharacteristic
438C22#193311yescharacteristic
439C22#193511yescharacteristic
440C22#200011yescharacteristic
441C22#200111yescharacteristic
442C22#206611yescharacteristic
443C22#206811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—162no—
456C4#181412yes—
457C4#186311yescharacteristic
458C4#186511yescharacteristic
459C4#187711yescharacteristic
460C4#187811yescharacteristic
461C4#202411yescharacteristic
462C4#202611yescharacteristic
463C4#204211yescharacteristic
464C4#204311yescharacteristic
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2#1106711yescharacteristic
472C2#1110411yescharacteristic
473C2#1111911yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#213292
#211692
#175752
C2×#14812
#80952
C2×#13012
#79242