Finite Group Databasebeta

← #279572

Subgroups — #279572

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29998C211yesmaximal, characteristic
2#29207C211yesmaximal, characteristic, complemented
3#22651C211yesmaximal, characteristic
4#22146C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10602C211yesmaximal, characteristic, complemented
8#10602C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9601C211yesmaximal, characteristic, complemented
11#9225C211yesmaximal, characteristic
12#9217C211yesmaximal, characteristic
13#8650C211yesmaximal, characteristic
14#8644C211yesmaximal, characteristic
15#8586C211yesmaximal, characteristic, complemented
16C2×#461C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#424—21no—
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2293C2211yescharacteristic
23#2088—21no—
24#2088—21no—
25#2065—21no—
26#2065—21no—
27#2059—21no—
28#2059—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1981C2211yescharacteristic
32#1968—21no—
33#1949—21no—
34#1949—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1933—21no—
41#1880—21no—
42#1865—21no—
43#1851—21no—
44#1850—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1798—21no—
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C22×M16—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×C4×Q8—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#461—22no—
76#461Q814yes—
77#436—22no—
78#436—41no—
79C8×D4—22no—
80C8×D4—41no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88#424—44no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390D411yescharacteristic
116#390D411yescharacteristic
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389D411yescharacteristic
124#389D411yescharacteristic
125#388—21no—
126#388—21no—
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
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—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151#335—41no—
152#335—41no—
153#332—41no—
154#332—41no—
155C22×Q8—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C2×SD16—24no—
159C2×SD16—81no—
160C2×SD16—81no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—81no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—81no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C2×Q812yes—
181#125—81no—
182#125—81no—
183C4×Q8—81no—
184C4×Q8—81no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C2×Q812yes—
188C4×D4#4112yes—
189#117—42no—
190#117—42no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224#107—81no—
225#107—81no—
226#106—81no—
227#106—81no—
228#106—81no—
229#106—81no—
230#106—81no—
231C4⋊C8—22no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
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—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248C4⋊C8—81no—
249C4⋊C8SD1614yescomplemented
250C4⋊C8C2×Q812yes—
251#103—24no—
252#103—81no—
253#103—81no—
254#102—81no—
255#102—81no—
256#102—81no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—81no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24—21no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4C22⋊Q812yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304SD16—84no—
305M16—84no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
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—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—84no—
335C2×C8C2×SD1612yes—
336C2×C8C2×SD1612yes—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
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—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4C22⋊Q812yes—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—81no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—82no—
384C42C2×SD1612yes—
385C42#13712yes—
386C42C22×Q811yescharacteristic
387C42C2×#4111yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C422−1+411yescharacteristic
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C2×#12111yescharacteristic
398Q8—42no—
399Q8—42no—
400Q8—81no—
401Q8—81no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
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#46112yes—
444C2×C4#46112yes—
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4#53711yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4D4×Q811yescharacteristic
449C2×C4#55211yescharacteristic
450C2×C4#55411yescharacteristic
451C2×C4C22×SD1611yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C8—22no—
460C8—22no—
461C8—41no—
462C8—41no—
463C8—42no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#241711yescharacteristic
477C22C2×#46111yescharacteristic
478C22#304511yescharacteristic
479C22#306111yescharacteristic
480C22#306211yescharacteristic
481C22#306711yescharacteristic
482C22#306911yescharacteristic
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4#308111yescharacteristic
497C4#308311yescharacteristic
498C4#325811yescharacteristic
499C4#325911yescharacteristic
500C4#329511yescharacteristic
501C4#333111yescharacteristic
502C4Q8×SD1611yescharacteristic
503C4#338511yescharacteristic
504C2—82no—
505C2#1919511yescharacteristic
506C2#1923211yescharacteristic
507C2#1954311yescharacteristic
508C2#1957811yescharacteristic
509C2#2481511yescharacteristic
510C2#2484811yescharacteristic
511C2#2486011yescharacteristic
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. #29207C2semidirect
5. #10654C2semidirect
6. #10652C2semidirect
7. #10602C2semidirect
8. #10602C2semidirect
9. #9900C2semidirect
10. #9601C2semidirect
15. #8586C2semidirect
249. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#299982
#292072
#226512
#221462
#106542
#106522
#106022
#99002
#96012
#92252
#92172
#86502
#86442
#85862