Finite Group Databasebeta

← #228867

Subgroups — #228867

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23248C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9599C211yesmaximal, characteristic, complemented
7#9594C211yesmaximal, characteristic, complemented
8#9537C211yesmaximal, characteristic, complemented
9#9536C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic
11#8843C211yesmaximal, characteristic
12#8262C211yesmaximal, characteristic, complemented
13#8204C211yesmaximal, characteristic
14C8×#102C211yesmaximal, characteristic
15#8172C211yesmaximal, characteristic
16C2×#474C411yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1988C411yescharacteristic
22#1985—21no—
23#1984—21no—
24#1947C411yescharacteristic
25#1940—21no—
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1918C411yescharacteristic
31#1918C411yescharacteristic
32#1862C411yescharacteristic
33#1862C411yescharacteristic
34#1850—21no—
35#1850—21no—
36#1831C2211yescharacteristic
37#1802C411yescharacteristic
38#1797—21no—
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41#1771—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C2×#336—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#474—24no—
76C8×D4—22no—
77C8×D4—22no—
78C8×D4—41no—
79C8×D4—41no—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#389—21no—
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—41no—
150#334—41no—
151D4⋊C8—22no—
152D4⋊C8Q814yescomplemented
153#322—41no—
154C82—41no—
155C22×Q8—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—22no—
181C4⋊Q8—41no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4C2×Q812yes—
191C4×D4#4112yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—81no—
213C2×C42—81no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—22no—
222#107—22no—
223#107—22no—
224#107—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8C2×Q812yes—
241#102—22no—
242#102—22no—
243#102—22no—
244#102M1614yes—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—24no—
248#97—42no—
249#97—42no—
250#97—81no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—81no—
263C4×C8C2×Q812yes—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271C24—21no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—81no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C4×Q812yes—
279C2×D4C2×M1612yes—
280C2×D4#13112yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—84no—
348C2×C8—84no—
349C2×C8C2×M1612yes—
350C2×C8#13712yes—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4C4×Q812yes—
368C4⋊C4C2×M1612yes—
369C4⋊C4#13112yes—
370C22⋊C4—24no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—42no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—22no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—81no—
391C42—81no—
392C42C22×Q811yescharacteristic
393C42C2×#4111yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C422−1+411yescharacteristic
396C23—22no—
397C23—22no—
398C23—22no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C2×C4×D411yescharacteristic
404Q8—84no—
405D4—22no—
406D4—22no—
407D4—22no—
408D4#44614yes—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4#44312yes—
450C2×C4C2×C4×Q811yescharacteristic
451C2×C4#52011yescharacteristic
452C2×C4#54611yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4#55211yescharacteristic
455C2×C4#55411yescharacteristic
456C2×C4C22×M1611yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57611yescharacteristic
464C8—24no—
465C8—22no—
466C8—22no—
467C8—42no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—24no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22#236011yescharacteristic
486C22#292911yescharacteristic
487C22D4×M1611yescharacteristic
488C22C4×#13711yescharacteristic
489C22#294811yescharacteristic
490C22C2×#44311yescharacteristic
491C22#295611yescharacteristic
492C4—22no—
493C4—21no—
494C4—21no—
495C4—41no—
496C4—42no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—41no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4C2×#44611yescharacteristic
506C4#296211yescharacteristic
507C4#299011yescharacteristic
508C4#299211yescharacteristic
509C4#331411yescharacteristic
510C4#331911yescharacteristic
511C4#338511yescharacteristic
512C4#338911yescharacteristic
513C2—44no—
514C2#1791311yescharacteristic
515C2#1792211yescharacteristic
516C2#1804711yescharacteristic
517C2#1805411yescharacteristic
518C2#2295411yescharacteristic
519C2#2297911yescharacteristic
520C2#2298211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23248C2semidirect
4. #9877C2semidirect
5. #9606C2semidirect
6. #9599C2semidirect
7. #9594C2semidirect
8. #9537C2semidirect
9. #9536C2semidirect
12. #8262C2semidirect
152. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#232482
#226962
C4×#4742
#98772
#96062
#95992
#95942
#95372
#95362
#88452
#88432
#82622
#82042
C8×#1022
#81722