Finite Group Databasebeta

← #498463

Subgroups — #498463

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33579C211yesmaximal, characteristic
2#23179C211yesmaximal, characteristic, complemented
3#22802C211yesmaximal, characteristic, complemented
4#13242C211yesmaximal, characteristic, complemented
5#13241C211yesmaximal, characteristic, complemented
6#13209C211yesmaximal, characteristic, complemented
7#13179C211yesmaximal, characteristic, complemented
8#13134C211yesmaximal, characteristic, complemented
9#13134C211yesmaximal, characteristic, complemented
10#13011C211yesmaximal, characteristic, complemented
11#13011C211yesmaximal, characteristic, complemented
12#12663C211yesmaximal, characteristic, complemented
13#12642C211yesmaximal, characteristic, complemented
14#12641C211yesmaximal, characteristic, complemented
15#12640C211yesmaximal, characteristic, complemented
16C2×#508C2211yescharacteristic
17C2×#508C2211yescharacteristic
18#3147C2211yescharacteristic
19C2×#443C2211yescharacteristic
20C2×#443C2211yescharacteristic
21C4×#135C2211yescharacteristic
22#2935—21no—
23#2925—21no—
24#2211C411yescharacteristic, complemented
25#2211C411yescharacteristic, complemented
26#2210C411yescharacteristic, complemented
27#2210C411yescharacteristic, complemented
28#2209C411yescharacteristic, complemented
29#2208C411yescharacteristic, complemented
30#2207C411yescharacteristic, complemented
31#2206C411yescharacteristic, complemented
32C2×#368C2211yescharacteristic
33C2×#368C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×#359C2211yescharacteristic
36C2×#359C2211yescharacteristic
37C2×#359C2211yescharacteristic
38C2×#359C2211yescharacteristic
39C2×#359C2211yescharacteristic
40C2×#359C2211yescharacteristic
41#2138C2211yescharacteristic
42#2138C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×#346C2211yescharacteristic
45C2×C4×C16C2211yescharacteristic
46#1980—21no—
47#1974C2211yescharacteristic
48#1974C2211yescharacteristic
49#1974C2211yescharacteristic
50#1974C2211yescharacteristic
51#1929—21no—
52#1929—21no—
53#1876—21no—
54#1876—21no—
55#1848C2211yescharacteristic
56#1806—21no—
57C4×#108C2211yescharacteristic
58#1772C2211yescharacteristic
59#1772C2211yescharacteristic
60C8×C4⋊C4C2211yescharacteristic
61#1382C2211yescharacteristic
62#1382C2211yescharacteristic
63#1381C2211yescharacteristic
64#1381C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135—21no—
68C2×#135C2311yescharacteristic
69C2×#135C2311yescharacteristic
70C2×#135C2311yescharacteristic
71C2×#131—21no—
72C2×#131—21no—
73#532—21no—
74C4×#41—21no—
75#508—41no—
76#508—41no—
77#508—41no—
78#508—41no—
79C2×Q32C2×C411yescharacteristic
80C2×Q32C2×C411yescharacteristic
81C2×SD32C2×C411yescharacteristic
82C2×SD32C2×C411yescharacteristic
83C2×SD32C2×C411yescharacteristic
84C2×SD32C2×C411yescharacteristic
85C2×D16C2×C411yescharacteristic
86C2×D16C2×C411yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91#495—21no—
92#495C2×C411yescharacteristic
93#495C2×C411yescharacteristic
94#494C2×C411yescharacteristic
95#493C2×C411yescharacteristic
96#492—21no—
97#443—22no—
98#443—22no—
99#443—41no—
100#443—41no—
101C4×Q16C2×C411yescharacteristic
102C4×Q16C2×C411yescharacteristic
103C4×SD16—21no—
104C4×SD16—21no—
105C4×D8C2×C411yescharacteristic
106C4×D8C2×C411yescharacteristic
107#436—41no—
108#435—41no—
109#435—41no—
110C8×D4—41no—
111#433—41no—
112#433—41no—
113#432—41no—
114#431—41no—
115C2×#108—21no—
116C2×#108C2311yescharacteristic
117C2×#108C2311yescharacteristic
118C2×#108C2311yescharacteristic
119C2×#108C2311yescharacteristic
120#427C2311yescharacteristic
121#423—21no—
122C2×C4⋊C8D411yescharacteristic
123C2×#104—21no—
124C2×#104—21no—
125C2×#104—21no—
126C2×#104—21no—
127C2×#104—21no—
128C2×#104—21no—
129#416—21no—
130#408—21no—
131#408—21no—
132#408—21no—
133#408—21no—
134#405—21no—
135#405—21no—
136C4×M16—21no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141#368—41no—
142#368—41no—
143C4⋊C16C2×C411yescharacteristic
144C4⋊C16C2×C411yescharacteristic
145C4⋊C16C2×C411yescharacteristic
146C4⋊C16C2×C411yescharacteristic
147#359—22no—
148#359—22no—
149#359—22no—
150#359—22no—
151#359—41no—
152#359—41no—
153#358C2×C411yescharacteristic
154#358C2×C411yescharacteristic
155#358C2×C411yescharacteristic
156#358C2×C411yescharacteristic
157#357C2×C411yescharacteristic
158#357C2×C411yescharacteristic
159#357C2×C411yescharacteristic
160#357C2×C411yescharacteristic
161#346—21no—
162#346—21no—
163#346—41no—
164C4×C16—41no—
165#341—21no—
166#339—21no—
167#339—21no—
168#336D411yescharacteristic
169#336D411yescharacteristic
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173#135—21no—
174#135—21no—
175#135—21no—
176#135—21no—
177#135—21no—
178#135—21no—
179#135—21no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184#135—21no—
185#135—81no—
186#135—81no—
187C2×Q16—21no—
188C2×Q16C22×C411yescharacteristic
189C2×Q16C22×C411yescharacteristic
190C2×Q16C22×C411yescharacteristic
191C2×SD16—21no—
192C2×SD16—21no—
193C2×SD16—21no—
194C2×SD16—21no—
195C2×SD16—21no—
196C2×D8—21no—
197C2×D8C22×C411yescharacteristic
198C2×D8C22×C411yescharacteristic
199C2×D8C22×C411yescharacteristic
200#131—42no—
201#131—42no—
202#131—81no—
203#131—81no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—41no—
209C2×M16—41no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2411yescharacteristic
216C4⋊Q8—21no—
217#127—21no—
218#124—21no—
219#124—21no—
220#121—41no—
221#121—41no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230#117—21no—
231#117—41no—
232C2×C4⋊C4—21no—
233C2×C42—41no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237Q32—41no—
238Q32—41no—
239SD32—41no—
240SD32—41no—
241SD32—41no—
242SD32—41no—
243D16—41no—
244D16—41no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16C22×C411yescharacteristic
255C2×C16C22×C411yescharacteristic
256C2×C16C22×C411yescharacteristic
257C2×C16C22×C411yescharacteristic
258C2×C16#4112yes—
259#108—21no—
260#108—21no—
261#108—21no—
262#108—21no—
263#108—21no—
264#108—21no—
265#108—21no—
266#108—81no—
267#108#4112yes—
268#107C22×C411yescharacteristic
269#107C22×C411yescharacteristic
270#106—21no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8D811yescharacteristic
274C4⋊C8D811yescharacteristic
275C4⋊C8D811yescharacteristic
276C4⋊C8D811yescharacteristic
277#104—42no—
278#104—42no—
279#104—81no—
280#104—81no—
281#104—81no—
282#104—81no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287C22⋊C8—41no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295#97—41no—
296#97—41no—
297#97—41no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—41no—
303C4×C8—41no—
304C4×C8—41no—
305C4×C8C22×C411yescharacteristic
306C4×C8C22×C411yescharacteristic
307C4×C8#4111yescharacteristic
308C4×C8#4111yescharacteristic
309#95—21no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4#13511yescharacteristic
341C22×C4#13511yescharacteristic
342C22×C4C22×D411yescharacteristic
343Q16—21no—
344Q16—21no—
345Q16—21no—
346Q16—21no—
347Q16—21no—
348Q16—21no—
349Q16—81no—
350SD16—41no—
351SD16—41no—
352SD16—41no—
353SD16—41no—
354SD16—41no—
355SD16—41no—
356SD16—81no—
357SD16—81no—
358D8—21no—
359D8—21no—
360D8—21no—
361D8—21no—
362D8—21no—
363D8—21no—
364D8—81no—
365M16—22no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—82no—
374M16—82no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—82no—
386C2×C8—82no—
387C2×C8C4×D411yescharacteristic
388C2×C8C4×D411yescharacteristic
389C2×C8C2×D811yescharacteristic
390C2×C8C2×D811yescharacteristic
391C2×C8C2×D811yescharacteristic
392C2×C8C2×D811yescharacteristic
393C2×C8C23×C411yescharacteristic
394C2×C8C2×#4111yescharacteristic
395C2×C8C2×#4111yescharacteristic
396C2×C8C2×#4111yescharacteristic
397C2×C8H(5,𝔽2)11yescharacteristic
398C2×C8H(5,𝔽2)11yescharacteristic
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C2×D811yescharacteristic
423C42C2×D811yescharacteristic
424C16—21no—
425C16—21no—
426C16—21no—
427C16—21no—
428C16—22no—
429C16—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23C2×#13511yescharacteristic
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—82no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—81no—
452D4—81no—
453D4—161no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—161no—
475C2×C4C4×D811yescharacteristic
476C2×C4C4×D811yescharacteristic
477C2×C4C4×D811yescharacteristic
478C2×C4C4×D811yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#55011yescharacteristic
482C2×C4C22×D811yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57711yescharacteristic
485C8—21no—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—21no—
491C8—42no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8C4×#4111yescharacteristic
496C8#51811yescharacteristic
497C8#55311yescharacteristic
498C8#55511yescharacteristic
499C22—21no—
500C22—21no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22C2×C4×D811yescharacteristic
511C22#330511yescharacteristic
512C22#339411yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4#236311yescharacteristic
525C4#294611yescharacteristic
526C4#329411yescharacteristic
527C4#339111yescharacteristic
528C2—21no—
529C2—21no—
530C2—161no—
531C2—161no—
532C2—161no—
533C2#2309311yescharacteristic
534C2#3346311yescharacteristic
535C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#33579C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23179C2semidirect
3. #22802C2semidirect
4. #13242C2semidirect
5. #13241C2semidirect
6. #13209C2semidirect
7. #13179C2semidirect
8. #13134C2semidirect
9. #13134C2semidirect
10. #13011C2semidirect
11. #13011C2semidirect
12. #12663C2semidirect
13. #12642C2semidirect
14. #12641C2semidirect
15. #12640C2semidirect
24. #2211C4semidirect
25. #2211C4semidirect
26. #2210C4semidirect
27. #2210C4semidirect
28. #2209C4semidirect
29. #2208C4semidirect
30. #2207C4semidirect
31. #2206C4semidirect

Maximal subgroups

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

subgroupindex
#335792
#231792
#228022
#132422
#132412
#132092
#131792
#131342
#130112
#126632
#126422
#126412
#126402