Finite Group Databasebeta

← #325069

Subgroups — #325069

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30203C211yesmaximal, characteristic
2#29354C211yesmaximal, characteristic
3#28687C211yesmaximal, characteristic, complemented
4#22309C211yesmaximal, characteristic, complemented
5#10601C211yesmaximal, characteristic, complemented
6#10599C211yesmaximal, characteristic
7#10597C211yesmaximal, characteristic, complemented
8#10595C211yesmaximal, characteristic
9#9903C211yesmaximal, characteristic, complemented
10#9699C211yesmaximal, characteristic
11#9575C211yesmaximal, characteristic, complemented
12#9572C211yesmaximal, characteristic
13#9567C211yesmaximal, characteristic, complemented
14#9564C211yesmaximal, characteristic
15#8279C211yesmaximal, characteristic, complemented
16C2×#498—21no—
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2084—21no—
21#2084—21no—
22#2071—21no—
23#2071—21no—
24#2071—21no—
25#2023—21no—
26#2023—21no—
27#2023—21no—
28#1990—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1985—21no—
34#1984—21no—
35#1982C2211yescharacteristic
36#1982C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1948C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1939C2211yescharacteristic
44#1939C2211yescharacteristic
45#1936—21no—
46#1936—21no—
47#1917—21no—
48#1916—21no—
49#1831C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×#124—21no—
74C2×#124—21no—
75C2×C4×Q8—21no—
76#498—44no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389D411yescharacteristic
110#389D411yescharacteristic
111#384—21no—
112#384—21no—
113#384—21no—
114#384D411yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C43C2311yescharacteristic
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#334—22no—
147#333—22no—
148#333—22no—
149#331—22no—
150#331—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—41no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—41no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—81no—
176C4⋊Q8—81no—
177C4⋊Q8—82no—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#124—82no—
183#124—82no—
184C4×Q8—81no—
185C4×Q8—81no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#106—42no—
220#106—42no—
221#106—42no—
222#106—42no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8#4112yes—
236C4⋊C8#4112yes—
237#102—42no—
238#102—42no—
239#102—81no—
240#102—81no—
241#102—81no—
242#102—81no—
243#102—81no—
244#102—81no—
245#97—81no—
246#97—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8—81no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257C24—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—81no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8—82no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—81no—
283C2×D4—81no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
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—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4AGL(1,ℤ/8)12yes—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C422−1+411yescharacteristic
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23C2×#12111yescharacteristic
388Q8—84no—
389Q8—161no—
390Q8—161no—
391Q8—164no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—161no—
401D4—161no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
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—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4#47112yes—
436C2×C4#47212yes—
437C2×C4#53211yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#54811yescharacteristic
440C2×C4#54811yescharacteristic
441C2×C4#55611yescharacteristic
442C2×C4#55611yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—42no—
452C8—42no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22#254511yescharacteristic
469C22#304211yescharacteristic
470C22#304311yescharacteristic
471C22#305511yescharacteristic
472C22#306311yescharacteristic
473C22C2×#47111yescharacteristic
474C22C2×#47211yescharacteristic
475C4—22no—
476C4—22no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—41no—
481C4—41no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4#317311yescharacteristic
489C4#317311yescharacteristic
490C4#324411yescharacteristic
491C4#324411yescharacteristic
492C4#336711yescharacteristic
493C4#337111yescharacteristic
494C4#337311yescharacteristic
495C4#337711yescharacteristic
496C2—321no—
497C2#1946211yescharacteristic
498C2#1947011yescharacteristic
499C2#1956111yescharacteristic
500C2#1956511yescharacteristic
501C2#2864911yescharacteristic
502C2#2866211yescharacteristic
503C2#2867011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28687C2semidirect
4. #22309C2semidirect
5. #10601C2semidirect
7. #10597C2semidirect
9. #9903C2semidirect
11. #9575C2semidirect
13. #9567C2semidirect
15. #8279C2semidirect

Maximal subgroups

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

subgroupindex
#302032
#293542
#286872
#223092
#106012
#105992
#105972
#105952
#99032
#96992
#95752
#95722
#95672
#95642
#82792