Finite Group Databasebeta

← #345720

Subgroups — #345720

Order 512.

613 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#26464C211yesmaximal, characteristic
2#26460C211yesmaximal, characteristic
3#26446C211yesmaximal, characteristic, complemented
4#26442C211yesmaximal, characteristic, complemented
5#10597C211yesmaximal, characteristic, complemented
6#10595C211yesmaximal, characteristic
7#10589C211yesmaximal, characteristic, complemented
8#10587C211yesmaximal, characteristic
9#10575C211yesmaximal, characteristic, complemented
10#10575C211yesmaximal, characteristic, complemented
11#9829C211yesmaximal, characteristic
12#9597C211yesmaximal, characteristic, complemented
13#9595C211yesmaximal, characteristic, complemented
14#9064C211yesmaximal, characteristic
15#9063C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C2×#476—21no—
18C2×#474—21no—
19#2551C2211yescharacteristic
20#2551C2211yescharacteristic
21#2369C2211yescharacteristic
22#2366C2211yescharacteristic
23#2088—21no—
24#2087—21no—
25#2067—21no—
26#2065—21no—
27#2058—21no—
28#2057—21no—
29#2023—21no—
30#2023—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1988—21no—
34#1987—21no—
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1982C2211yescharacteristic
38#1982C2211yescharacteristic
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1947C2211yescharacteristic
44#1947C2211yescharacteristic
45#1939C2211yescharacteristic
46#1939C2211yescharacteristic
47#1939C2211yescharacteristic
48#1939C2211yescharacteristic
49#1921—21no—
50#1919—21no—
51#1864—21no—
52#1862—21no—
53#1834C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C4×#106C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C22×SD16—41no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#492—41no—
77#492Q814yescomplemented
78#476—44no—
79#474—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107D411yescharacteristic
83C2×#107D411yescharacteristic
84C2×#106—21no—
85C2×#106—21no—
86C2×#106D411yescharacteristic
87C2×#106D411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#395—41no—
108#395—41no—
109#391—41no—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#386—41no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—41no—
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#334—41no—
149#334—41no—
150#333—41no—
151#331—41no—
152#322—41no—
153C22×Q8—21no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C23×C4—41no—
160C2×SD16—44no—
161C2×SD16—82no—
162C2×SD16—82no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—81no—
180C4⋊Q8—81no—
181C4⋊Q8—81no—
182C4⋊Q8—81no—
183C4⋊Q8C2×Q812yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C2×Q812yes—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—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—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
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×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—41no—
221C2×C42C2411yescharacteristic
222#107—41no—
223#107—42no—
224#107—81no—
225#107SD1614yescomplemented
226#106—22no—
227#106—22no—
228#106—42no—
229#106—81no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242#102—42no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—81no—
249#102—81no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8C2×Q812yes—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—41no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—161no—
274C2×Q8—161no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—161no—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308SD16—88no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
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—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—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8C2×SD1612yes—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
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—82no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—161no—
367C4⋊C4C2×SD1612yes—
368C4⋊C4C2×SD1612yes—
369C4⋊C4AGL(1,ℤ/8)12yes—
370C4⋊C4#13712yes—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—161no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—161no—
383C42C22×Q811yescharacteristic
384C42H(5,𝔽2)11yescharacteristic
385C422−1+411yescharacteristic
386C422−1+411yescharacteristic
387C23—42no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23D4211yescharacteristic
395Q8—82no—
396Q8—161no—
397Q8—164no—
398Q8—164no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—82no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—161no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
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—42no—
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—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—164no—
445C2×C4#53711yescharacteristic
446C2×C4#53711yescharacteristic
447C2×C4#54311yescharacteristic
448C2×C4D4×Q811yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4#56311yescharacteristic
451C2×C4C22×SD1611yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C8—22no—
460C8—42no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22#268311yescharacteristic
476C22#326411yescharacteristic
477C22#327011yescharacteristic
478C22#327511yescharacteristic
479C22D4×SD1611yescharacteristic
480C22#328811yescharacteristic
481C22#328811yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—161no—
494C4—161no—
495C4#324611yescharacteristic
496C4#325411yescharacteristic
497C4#326211yescharacteristic
498C4#326311yescharacteristic
499C4Q8×SD1611yescharacteristic
500C4#338511yescharacteristic
501C4#338611yescharacteristic
502C4#338811yescharacteristic
503C2—162no—
504C2#1976311yescharacteristic
505C2#1980911yescharacteristic
506C2#2016611yescharacteristic
507C2#2020911yescharacteristic
508C2#3124011yescharacteristic
509C2#3125211yescharacteristic
510C2#3125811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×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
3. #26446C2semidirect
4. #26442C2semidirect
5. #10597C2semidirect
7. #10589C2semidirect
9. #10575C2semidirect
10. #10575C2semidirect
12. #9597C2semidirect
13. #9595C2semidirect
77. #492Q8semidirect
225. #107SD16semidirect

Maximal subgroups

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

subgroupindex
#264642
#264602
#264462
#264422
#105972
#105952
#105892
#105872
#105752
#98292
#95972
#95952
#90642
#90632