Finite Group Databasebeta

← #229320

Subgroups — #229320

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23221C211yesmaximal, characteristic, complemented
2#23061C211yesmaximal, characteristic
3#23060C211yesmaximal, characteristic
4#22707C211yesmaximal, characteristic
5C2×#1683C211yesmaximal, characteristic
6#9840C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9566C211yesmaximal, characteristic, complemented
10#8984C211yesmaximal, characteristic
11#8976C211yesmaximal, characteristic
12#8929C211yesmaximal, characteristic, complemented
13#8926C211yesmaximal, characteristic, complemented
14#8285C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#474C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2369C2211yescharacteristic
21#2365C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1981C2211yescharacteristic
29#1956C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1935—21no—
35#1934—21no—
36#1934—21no—
37#1927—21no—
38#1926—21no—
39#1925—21no—
40#1920—21no—
41#1918—21no—
42#1865—21no—
43#1854—21no—
44#1854—21no—
45#1834C2211yescharacteristic
46#1808—21no—
47C4×#107—21no—
48C4×#106—21no—
49C4×#106C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1763—21no—
54C4×#102C2211yescharacteristic
55#1735—21no—
56#1683C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75#487—22no—
76#487C2×C414yes—
77#476—22no—
78#476C2×C414yes—
79#474—22no—
80#474C2×C414yes—
81C8×D4—22no—
82C8×D4C2×C414yes—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106D411yescharacteristic
90C2×#106D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—41no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102D411yescharacteristic
103C2×#102D411yescharacteristic
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#391—41no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#388—41no—
119#388—41no—
120#387—41no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—41no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yes—
154#333C2×C414yes—
155#332C2×C414yes—
156D4⋊C8—22no—
157#322—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—81no—
169C22×C8—81no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4×Q8—42no—
183C4×Q8—81no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C22×C412yes—
187C4×D4#4112yes—
188#117—42no—
189#117—42no—
190#117—81no—
191#117—81no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2411yescharacteristic
224#107—24no—
225#107—42no—
226#107—81no—
227#107—81no—
228#106—24no—
229#106—22no—
230#106—22no—
231#106—42no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240#103—24no—
241#103—24no—
242#103—42no—
243#103—81no—
244#103—82no—
245#102—24no—
246#102—24no—
247#102—41no—
248#102—41no—
249C22⋊C8—24no—
250#97—81no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8C22×C412yes—
258C4×C8#4112yes—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4C4×D412yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—84no—
331C2×C8C4×D412yes—
332C2×C8C4×D412yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—82no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—161no—
364C4⋊C4C4×D412yes—
365C4⋊C4#13512yes—
366C4⋊C4#13512yes—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—161no—
372C22⋊C4—161no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—82no—
380C42—82no—
381C42—82no—
382C42—161no—
383C42C23×C411yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C422−1+411yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23D4211yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—82no—
398Q8—81no—
399Q8—81no—
400Q8—161no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
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—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4#52011yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54811yescharacteristic
449C2×C4#55311yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57511yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57811yescharacteristic
457C2×C4#57811yescharacteristic
458C8—22no—
459C8—22no—
460C8—41no—
461C8—41no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—81no—
466C8—81no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#236111yescharacteristic
475C22#326911yescharacteristic
476C22#326911yescharacteristic
477C22#328011yescharacteristic
478C22#328211yescharacteristic
479C22#328811yescharacteristic
480C22#328911yescharacteristic
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—82no—
492C4—161no—
493C4—161no—
494C4C4×#13511yescharacteristic
495C4#294311yescharacteristic
496C4#294611yescharacteristic
497C4#295011yescharacteristic
498C4#334911yescharacteristic
499C4#334911yescharacteristic
500C4#336011yescharacteristic
501C4#336111yescharacteristic
502C2—82no—
503C2#1977811yescharacteristic
504C2#1978411yescharacteristic
505C2#1979411yescharacteristic
506C2#1980011yescharacteristic
507C2#2299911yescharacteristic
508C2#2303011yescharacteristic
509C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23221C2semidirect
6. #9840C2semidirect
7. #9619C2semidirect
8. #9568C2semidirect
9. #9566C2semidirect
12. #8929C2semidirect
13. #8926C2semidirect
14. #8285C2semidirect
56. #1683C4semidirect

Maximal subgroups

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

subgroupindex
#232212
#230612
#230602
#227072
C2×#16832
#98402
#96192
#95682
#95662
#89842
#89762
#89292
#89262
#82852
#82022