Finite Group Databasebeta

← #243279

Subgroups — #243279

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21987C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21973C211yesmaximal, characteristic
4#21962C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10655C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10654C211yesmaximal, characteristic, complemented
10#10654C211yesmaximal, characteristic, complemented
11#10653C211yesmaximal, characteristic, complemented
12#10653C211yesmaximal, characteristic, complemented
13#8044C211yesmaximal, characteristic
14C2×#1296C211yesmaximal, characteristic
15#7879C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2283C2211yescharacteristic
23#1949—21no—
24#1949—21no—
25#1948—21no—
26#1948—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1853—21no—
40#1853—21no—
41#1853—21no—
42#1852—21no—
43#1850—21no—
44#1850—21no—
45#1847—21no—
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1843—21no—
49#1836—21no—
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1832—21no—
53#1808—21no—
54#1807—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432Q814yes—
76#431—22no—
77#431—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8Q814yes—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×#97—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#384—21no—
115#384—21no—
116#384—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#383—21no—
122#382—21no—
123#382D411yescharacteristic
124#382D411yescharacteristic
125#381—21no—
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129#380D411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135#376—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—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—42no—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8—81no—
167C22×C8—81no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172#117—82no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4—81no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—81no—
212#107—81no—
213#107—81no—
214#107—81no—
215#106—81no—
216#106—81no—
217#106—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221#106—81no—
222#106—81no—
223C4⋊C8—24no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241C4⋊C8C2×Q812yes—
242C4⋊C8C2×Q812yes—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#97—24no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24#12111yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12312yes—
289C22×C4#12512yes—
290C22×C4#12512yes—
291C22×C4C4⋊Q812yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
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—84no—
326C2×C8—84no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
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—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—82no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—82no—
366C42C22⋊Q812yes—
367C42C22⋊Q812yes—
368C42#12312yes—
369C42#12512yes—
370C42#12512yes—
371C42C4⋊Q812yes—
372C42C22×Q811yescharacteristic
373C42C22×Q811yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23#39812yes—
381C23#47112yes—
382C23#47112yes—
383C23C2×#12111yescharacteristic
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—164no—
427C2×C4#39812yes—
428C2×C4#39812yes—
429C2×C4#39812yes—
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4C2×#12511yescharacteristic
434C2×C4C2×#12511yescharacteristic
435C2×C4C2×C4⋊Q811yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C2×C4#57811yescharacteristic
444C8—42no—
445C8—42no—
446C8—42no—
447C8—42no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—41no—
452C8—81no—
453C8—81no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22#209014yes—
461C22C2×#39811yescharacteristic
462C22#304511yescharacteristic
463C22#304611yescharacteristic
464C22#306211yescharacteristic
465C22#306311yescharacteristic
466C22C2×#47111yescharacteristic
467C22C2×#47111yescharacteristic
468C4—22no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#308111yescharacteristic
482C4#308211yescharacteristic
483C4#308311yescharacteristic
484C4#308411yescharacteristic
485C4#309611yescharacteristic
486C4#309711yescharacteristic
487C4#316311yescharacteristic
488C4#316411yescharacteristic
489C2—24no—
490C2#1958011yescharacteristic
491C2#1958111yescharacteristic
492C2#1958811yescharacteristic
493C2#1958911yescharacteristic
494C2#2366211yescharacteristic
495C2#2367111yescharacteristic
496C2C2×#209011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10656C2semidirect
6. #10656C2semidirect
7. #10655C2semidirect
8. #10655C2semidirect
9. #10654C2semidirect
10. #10654C2semidirect
11. #10653C2semidirect
12. #10653C2semidirect

Maximal subgroups

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

subgroupindex
#219872
#219812
#219732
#219622
#106562
#106552
#106542
#106532
#80442
C2×#12962
#78792