Finite Group Databasebeta

← #116406

Subgroups — #116406

Order 512.

704 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
1C2×#1994C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#10839C211yesmaximal, characteristic
6#8079C211yesmaximal, characteristic
7#7957C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C43C2211yescharacteristic
15#1994—21no—
16#1994—21no—
17#1994C412yescomplemented
18#1994C412yescomplemented
19#1842—21no—
20#1831—21no—
21#1797—22no—
22C4×C4⋊C8—22no—
23C2×#339—21no—
24C2×#339—21no—
25C2×#339—21no—
26C2×#339—21no—
27#1520—22no—
28#1518—22no—
29#1503—22no—
30#1493—22no—
31#1487—22no—
32#1481—22no—
33#1301—22no—
34#1286—22no—
35C22×M16—21no—
36C22×M16—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533C2×C412yes—
44#533D412yes—
45C2×C4⋊Q8Q811yescharacteristic
46C2×C4⋊Q8Q811yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8D412yes—
84C2×C4×C8—21no—
85C2×C4×C8D412yes—
86#384—21no—
87#382D412yes—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C43—21no—
93C43—21no—
94C43C2×C411yescharacteristic
95C43C2×C411yescharacteristic
96C43D411yescharacteristic
97C43D411yescharacteristic
98#339—22no—
99#339—22no—
100#339—22no—
101#339—22no—
102#339—42no—
103#339—42no—
104#336—24no—
105#328—28no—
106Q8⋊C8—28no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C22×C8—41no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C4⋊C412yes—
137C4⋊Q8C4⋊C412yes—
138#125—22no—
139#125—22no—
140#125—41no—
141#125—41no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—22no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—44no—
194C2×C42—44no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42SD1612yes—
202C2×C42Q1612yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—48no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227C4⋊C8—41no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—41no—
235C4×C8—41no—
236#95—22no—
237#95—22no—
238C24#12111yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—82no—
244C2×Q8—82no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12611yescharacteristic
280C22×C4C2×SD1611yescharacteristic
281C22×C4C2×Q1611yescharacteristic
282C22×C4#13511yescharacteristic
283C22×C4#13511yescharacteristic
284C22×C4AGL(1,ℤ/8)11yescharacteristic
285C22×C4#13711yescharacteristic
286C22×C4#13711yescharacteristic
287C22×C4#13711yescharacteristic
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—81no—
293M16—81no—
294M16—81no—
295M16—81no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—81no—
303C2×C8—81no—
304C2×C8—81no—
305C2×C8—81no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C42—24no—
327C42—24no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—22no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—41no—
346C42—81no—
347C42—81no—
348C42#10312yes—
349C42#10312yes—
350C42C2×C4⋊C411yescharacteristic
351C42#11711yescharacteristic
352C42C4×D411yescharacteristic
353C42C4×D411yescharacteristic
354C42#12011yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42C22⋊Q811yescharacteristic
358C42C22⋊Q811yescharacteristic
359C42C22⋊Q811yescharacteristic
360C42#12311yescharacteristic
361C42#12311yescharacteristic
362C42#12411yescharacteristic
363C42#12411yescharacteristic
364C42#12611yescharacteristic
365C42#12711yescharacteristic
366C23—41no—
367C23—41no—
368C23#38611yescharacteristic
369C23#39411yescharacteristic
370C23#39611yescharacteristic
371C23#46511yescharacteristic
372C23#46711yescharacteristic
373C23#46811yescharacteristic
374C23#47011yescharacteristic
375Q8—44no—
376Q8—82no—
377Q8—82no—
378Q8—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38011yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38511yescharacteristic
417C2×C4#38811yescharacteristic
418C2×C4#38811yescharacteristic
419C2×C4#39011yescharacteristic
420C2×C4#39211yescharacteristic
421C2×C4#39311yescharacteristic
422C2×C4#39411yescharacteristic
423C2×C4#39611yescharacteristic
424C2×C4#39711yescharacteristic
425C2×C4#39911yescharacteristic
426C2×C4C2×#10311yescharacteristic
427C2×C4#41611yescharacteristic
428C2×C4#41711yescharacteristic
429C2×C4#41911yescharacteristic
430C2×C4#46011yescharacteristic
431C2×C4#46111yescharacteristic
432C2×C4#46111yescharacteristic
433C2×C4#46211yescharacteristic
434C2×C4#46211yescharacteristic
435C2×C4#46211yescharacteristic
436C2×C4#46311yescharacteristic
437C2×C4#46311yescharacteristic
438C2×C4#46411yescharacteristic
439C2×C4#46411yescharacteristic
440C2×C4#46411yescharacteristic
441C2×C4#46411yescharacteristic
442C2×C4#46511yescharacteristic
443C2×C4#46711yescharacteristic
444C2×C4#46811yescharacteristic
445C2×C4#47011yescharacteristic
446C2×C4#47511yescharacteristic
447C2×C4#47711yescharacteristic
448C2×C4#47911yescharacteristic
449C2×C4#47911yescharacteristic
450C8—82no—
451C8—82no—
452C8—82no—
453C8—82no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22#143911yescharacteristic
465C22#189211yescharacteristic
466C22#189411yescharacteristic
467C22#205511yescharacteristic
468C22#205611yescharacteristic
469C22#206611yescharacteristic
470C22#206811yescharacteristic
471C4—24no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#180311yescharacteristic
485C4#180511yescharacteristic
486C4#186112yes—
487C4#198111yescharacteristic
488C4#198311yescharacteristic
489C4#203112yes—
490C4#205511yescharacteristic
491C4#205611yescharacteristic
492C4#206611yescharacteristic
493C4#206811yescharacteristic
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2#1107711yescharacteristic
501C2#1108011yescharacteristic
502C2#1108111yescharacteristic
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
17. #1994C4semidirect
18. #1994C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19942
#208082
#175752
#174862
#108392
#80792
#79572