Finite Group Databasebeta

← #118025

Subgroups — #118025

Order 512.

670 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20747C211yesmaximal, characteristic
2#20718C211yesmaximal, characteristic
3#17806C211yesmaximal, characteristic, complemented
4#17443C211yesmaximal, characteristic
5#7886C211yesmaximal, characteristic, complemented
6#7870C211yesmaximal, characteristic, complemented
7#7854C211yesmaximal, characteristic, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1826—21no—
16#1818—21no—
17#1806—22no—
18#1798—22no—
19#1789—21no—
20#1780—21no—
21C4×#104—22no—
22C4×C22⋊C8—22no—
23#1574—22no—
24#1573—22no—
25#1564—22no—
26#1488—22no—
27#1487C411yescharacteristic, complemented
28#1487C411yescharacteristic, complemented
29#1486C411yescharacteristic, complemented
30#1486C411yescharacteristic, complemented
31#1476—22no—
32#1278—22no—
33#1276—22no—
34#1274—22no—
35C22×#41D411yescharacteristic
36C4×#41—21no—
37C4×#41—21no—
38C4×#41—21no—
39C4×#41—21no—
40C4×#41—21no—
41C4×#41—21no—
42C4×#41—21no—
43C4×#41—21no—
44C2×C4×Q8C2×C411yescharacteristic
45C2×C4×Q8Q811yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×#117—21no—
51C2×#117—21no—
52C2×#117D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
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#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C411yescharacteristic
74#431C2×C411yescharacteristic
75#431D412yes—
76#423C2×C411yescharacteristic
77#423C2×C411yescharacteristic
78#423D412yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8D412yes—
82C2×#104—24no—
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C4×M16—21no—
90C4×M16—21no—
91C2×C4×C8—21no—
92C2×C4×C8D412yes—
93C2×C4×C8D412yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98#376—21no—
99C43D412yes—
100#339—24no—
101#336—24no—
102#335—24no—
103#334—24no—
104Q8⋊C8—22no—
105Q8⋊C8—22no—
106D4⋊C8—22no—
107D4⋊C8—22no—
108C2×#41—21no—
109C2×#41—21no—
110C2×#41—21no—
111C2×#41—21no—
112C2×#41—44no—
113C22×Q8C4⋊C411yescharacteristic
114C22×D4—21no—
115C22×D4C4⋊C411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4×Q8—21no—
129C4×Q8—21no—
130C4×Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152#117—24no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157#117—42no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C4⋊C411yescharacteristic
166C2×C4⋊C4C4⋊C411yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×Q811yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206#104—48no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—22no—
210C22⋊C8—48no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#97—41no—
220#97—41no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—41no—
225C4×C8—41no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8C22⋊C412yes—
229C4×C8C22⋊C412yes—
230#95—41no—
231#95—41no—
232C24—21no—
233C24#12311yescharacteristic
234#41—24no—
235#41—24no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—42no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C4⋊C411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4C22⋊Q811yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12611yescharacteristic
288C22×C4#12611yescharacteristic
289M16—44no—
290M16—42no—
291M16—42no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—44no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—24no—
319C42—42no—
320C42—44no—
321C42—44no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42C2×C22⋊C411yescharacteristic
329C42#11711yescharacteristic
330C42C4×D411yescharacteristic
331C42C4×D411yescharacteristic
332C42#12011yescharacteristic
333C42#12111yescharacteristic
334C42#12111yescharacteristic
335C42#12111yescharacteristic
336C42#12111yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42#12311yescharacteristic
342C42#12411yescharacteristic
343C42#12711yescharacteristic
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23#38511yescharacteristic
353C23#39411yescharacteristic
354C23#39711yescharacteristic
355Q8—22no—
356Q8—22no—
357Q8—41no—
358Q8—41no—
359D4—22no—
360D4—22no—
361D4—41no—
362D4—41no—
363D4—44no—
364D4—44no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#38011yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#39011yescharacteristic
407C2×C4#39211yescharacteristic
408C2×C4#39311yescharacteristic
409C2×C4#39411yescharacteristic
410C2×C4#39611yescharacteristic
411C2×C4#39611yescharacteristic
412C2×C4#39911yescharacteristic
413C2×C4#44312yes—
414C2×C4#44412yes—
415C8—42no—
416C8—42no—
417C8—82no—
418C8—81no—
419C8—81no—
420C8—82no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22—42no—
432C22—82no—
433C22—82no—
434C22—81no—
435C22—81no—
436C22#143911yescharacteristic
437C22#186612yes—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—84no—
448C4—81no—
449C4—81no—
450C4#192912yes—
451C4#197411yescharacteristic
452C4#197511yescharacteristic
453C4#201012yes—
454C4#209812yes—
455C2—22no—
456C2—41no—
457C2—41no—
458C2—82no—
459C2#1108411yescharacteristic
460C2#1110011yescharacteristic
461C2#1112511yescharacteristic
subgroupG/H
centerC42#127
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17806C2semidirect
5. #7886C2semidirect
6. #7870C2semidirect
7. #7854C2semidirect
27. #1487C4semidirect
28. #1487C4semidirect
29. #1486C4semidirect
30. #1486C4semidirect

Maximal subgroups

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

subgroupindex
#207472
#207182
#178062
#174432
#78862
#78702
#78542