Finite Group Databasebeta

← #279402

Subgroups — #279402

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33068C211yesmaximal, characteristic
2#29220C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic
4#21614C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#9907C211yesmaximal, characteristic, complemented
10#9606C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#9215C211yesmaximal, characteristic
13#8648C211yesmaximal, characteristic
14#8646C211yesmaximal, characteristic
15#8594C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#2101—21no—
23#2101—21no—
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1989—21no—
27#1989—21no—
28#1987C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1920—21no—
35#1920—21no—
36#1919—21no—
37#1864—21no—
38#1864—21no—
39#1862—21no—
40#1850—21no—
41#1850—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1797—21no—
50#1776—21no—
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1762—21no—
54C2×#336—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#476—22no—
77#476Q814yes—
78C8×D4—22no—
79C8×D4—22no—
80C8×D4—41no—
81C8×D4—41no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
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×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#391—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
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#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—41no—
152#333—41no—
153#333—41no—
154C22×Q8—21no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C2×Q812yes—
179#125—42no—
180#125—42no—
181C4×Q8—82no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—41no—
188C4×D4C2×Q812yes—
189C4×D4#4112yes—
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—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#106—24no—
225#106—42no—
226#106—42no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
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—41no—
239C4⋊C8—41no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8C2×Q812yes—
243#102—24no—
244#102—42no—
245#102—42no—
246#102—42no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C22⋊Q812yes—
270C2×D4#13512yes—
271C2×D4#13712yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
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—41no—
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—
328C2×C8—84no—
329C2×C8—84no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
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—41no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4C22⋊Q812yes—
357C4⋊C4#13512yes—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—42no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—81no—
375C42—81no—
376C42—82no—
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C42H(5,𝔽2)11yescharacteristic
380C422−1+411yescharacteristic
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C2×#12111yescharacteristic
389Q8—82no—
390Q8—82no—
391D4—22no—
392D4—22no—
393D4—22no—
394D4#47914yes—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—81no—
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—81no—
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4#53711yescharacteristic
438C2×C4#54611yescharacteristic
439C2×C4D4×Q811yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55411yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57511yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C8—42no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—24no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22#241711yescharacteristic
468C22#304011yescharacteristic
469C22#304211yescharacteristic
470C22#305011yescharacteristic
471C22#305611yescharacteristic
472C22#306711yescharacteristic
473C22#306811yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4—81no—
483C4—82no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4C2×#47911yescharacteristic
488C4#307611yescharacteristic
489C4#324811yescharacteristic
490C4#325011yescharacteristic
491C4#331111yescharacteristic
492C4#331111yescharacteristic
493C4#338811yescharacteristic
494C4#338811yescharacteristic
495C2—44no—
496C2#1927811yescharacteristic
497C2#1931811yescharacteristic
498C2#1939411yescharacteristic
499C2#1946611yescharacteristic
500C2#2483411yescharacteristic
501C2#2484011yescharacteristic
502C2#2485911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
2. #29220C2semidirect
5. #10657C2semidirect
6. #10655C2semidirect
7. #10601C2semidirect
8. #10597C2semidirect
9. #9907C2semidirect
10. #9606C2semidirect
15. #8594C2semidirect

Maximal subgroups

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

subgroupindex
#330682
#292202
#226962
#216142
#106572
#106552
#106012
#105972
#99072
#96062
#92152
#86482
#86462
#85942