Finite Group Databasebeta

← #228289

Subgroups — #228289

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23759C211yesmaximal, characteristic
2#23050C211yesmaximal, characteristic
3#22640C211yesmaximal, characteristic
4#9594C211yesmaximal, characteristic, complemented
5#9557C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9534C211yesmaximal, characteristic, complemented
8#9533C211yesmaximal, characteristic, complemented
9#9075C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic
12#8290C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14#8148C211yesmaximal, characteristic
15#7894C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2283C2211yescharacteristic
21#2101C411yescharacteristic
22#2097C411yescharacteristic
23#2085C411yescharacteristic
24#2077C411yescharacteristic
25#2075C411yescharacteristic
26#2074C411yescharacteristic
27#2036C411yescharacteristic
28#2035C411yescharacteristic
29#1984C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1924—21no—
34#1923—21no—
35#1923—21no—
36#1923—21no—
37#1916—21no—
38#1840—21no—
39#1797C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4—21no—
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1768—21no—
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊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
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C8×Q8—22no—
71C8×Q8—41no—
72#431—22no—
73#431—22no—
74#431—41no—
75#431—41no—
76C2×#107C2×C411yescharacteristic
77C2×#107C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C411yescharacteristic
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#387—21no—
114#385—21no—
115#383—21no—
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#381—21no—
119#380C2×C411yescharacteristic
120#380C2×C411yescharacteristic
121#380C2×C411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
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#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—41no—
154#322—22no—
155C82—41no—
156C22×Q8C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206#107—22no—
207#107—22no—
208#106—22no—
209#106—41no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8#4112yes—
223#103—22no—
224#103—22no—
225#103—22no—
226#103—41no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233C22⋊C8—81no—
234#97—81no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247C4×C8—42no—
248C4×C8—42no—
249C4×C8—81no—
250C4×C8#4112yes—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264C24C4×D411yescharacteristic
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8#13112yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8#13112yes—
341C2×C8#13512yes—
342C2×C8#13512yes—
343C4⋊C4—24no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4#13112yes—
361C22⋊C4—24no—
362C22⋊C4—42no—
363C22⋊C4—81no—
364C22⋊C4—81no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C42H(5,𝔽2)11yescharacteristic
382C422−1+411yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23C2×C4×D411yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C4×#4111yescharacteristic
441C2×C4#52011yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#55111yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4C2×#13111yescharacteristic
447C2×C4C2×#13111yescharacteristic
448C2×C4C2×#13111yescharacteristic
449C2×C4#56811yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57811yescharacteristic
454C8—22no—
455C8—22no—
456C8—22no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—42no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22#235911yescharacteristic
476C22#292911yescharacteristic
477C22#293511yescharacteristic
478C22C4×#13511yescharacteristic
479C22#294711yescharacteristic
480C22#295511yescharacteristic
481C22#295711yescharacteristic
482C4—21no—
483C4—21no—
484C4—41no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—42no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4#296611yescharacteristic
496C4#296911yescharacteristic
497C4#299211yescharacteristic
498C4#299411yescharacteristic
499C4#330111yescharacteristic
500C4#330211yescharacteristic
501C4#333711yescharacteristic
502C4#334211yescharacteristic
503C2—81no—
504C2#1790411yescharacteristic
505C2#1791911yescharacteristic
506C2#1806911yescharacteristic
507C2#1808111yescharacteristic
508C2#2290111yescharacteristic
509C2#2291911yescharacteristic
510C2#2294911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9594C2semidirect
5. #9557C2semidirect
6. #9548C2semidirect
7. #9534C2semidirect
8. #9533C2semidirect
10. #8846C2semidirect
12. #8290C2semidirect
13. C8×#103C2semidirect

Maximal subgroups

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

subgroupindex
#237592
#230502
#226402
#95942
#95572
#95482
#95342
#95332
#90752
#88462
#88312
#82902
C8×#1032
#81482
#78942