Finite Group Databasebeta

← #391673

Subgroups — #391673

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23622C211yesmaximal, characteristic
2#11375C211yesmaximal, characteristic, complemented
3#11367C211yesmaximal, characteristic, complemented
4#11363C211yesmaximal, characteristic, complemented
5#11294C211yesmaximal, characteristic, complemented
6#11254C211yesmaximal, characteristic, complemented
7#11251C211yesmaximal, characteristic, complemented
8#11165C211yesmaximal, characteristic
9#11165C211yesmaximal, characteristic
10#11159C211yesmaximal, characteristic
11#11146C211yesmaximal, characteristic
12#11141C211yesmaximal, characteristic
13#10955C211yesmaximal, characteristic
14#10906C211yesmaximal, characteristic, complemented
15#10901C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#430C2211yescharacteristic
20#2911C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C2×#421C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C22×#104C2211yescharacteristic
25C2×#413C2211yescharacteristic
26C2×#413C2211yescharacteristic
27C2×#413C2211yescharacteristic
28#2873C2211yescharacteristic
29#2070C414yescomplemented
30#2069C414yescomplemented
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37#1742C2211yescharacteristic
38#1742C2211yescharacteristic
39#1738C2211yescharacteristic
40#1738C2211yescharacteristic
41#1738C2211yescharacteristic
42#1738C2211yescharacteristic
43#1733C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#338C2211yescharacteristic
48C2×#338C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50#1728C2211yescharacteristic
51#1728C2211yescharacteristic
52#1728C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#430—41no—
69#430—41no—
70#430—41no—
71#430—41no—
72#430—41no—
73#430C2×C414yes—
74#430C2×C414yes—
75#430C2×C414yes—
76C2×#108—22no—
77C2×#108—41no—
78C2×#108—41no—
79C2×#108—41no—
80#428—41no—
81#428—41no—
82#427—41no—
83#421—22no—
84#421—41no—
85#421—41no—
86#421—41no—
87#421—41no—
88#421—41no—
89C2×#104—22no—
90C2×#104—41no—
91C2×#104—41no—
92C2×#104—41no—
93C2×#104—41no—
94C2×#104—41no—
95#413—41no—
96#413—41no—
97#413—41no—
98#413—41no—
99#413—41no—
100#413—41no—
101C2×#101C2×C412yes—
102C2×#101C2×C412yes—
103C2×#101C2×C412yes—
104C2×#100C2×C412yes—
105C2×#100C2×C412yes—
106C2×#100C2×C412yes—
107#405—41no—
108#405—41no—
109C4×M16—41no—
110#344—22no—
111#344—22no—
112#344—41no—
113#344—41no—
114#344—41no—
115#344—41no—
116#343—41no—
117#343—41no—
118#341—41no—
119#341—41no—
120#341—41no—
121#341—41no—
122#339—22no—
123#339—41no—
124#339—41no—
125#339—41no—
126#338—41no—
127#338C2×C414yes—
128H(3,ℤ/4)—41no—
129#336—41no—
130#336—41no—
131#336—41no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C22×Q8C22×C411yescharacteristic
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2411yescharacteristic
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16C22×C412yes—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C22×C412yes—
207C2×M16C22×C412yes—
208C2×M16C22×C412yes—
209C2×M16#4112yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117#4112yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247#108—44no—
248#108—44no—
249#108—44no—
250#108—44no—
251#107—42no—
252#106—42no—
253#104—44no—
254#104—44no—
255#104—44no—
256#104—44no—
257#101—42no—
258#101—42no—
259#101—42no—
260#100—42no—
261#100—42no—
262#100—42no—
263#97—42no—
264C4×C8—42no—
265C24—21no—
266C242−1+411yescharacteristic
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C23×C411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C42−1+411yescharacteristic
327C22×C42−1+411yescharacteristic
328C22×C42−1+411yescharacteristic
329M16—22no—
330M16—22no—
331M16—22no—
332M16—22no—
333M16—22no—
334M16—22no—
335M16—22no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—41no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C22⋊C4—22no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42—22no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—81no—
425C23—161no—
426C23#51911yescharacteristic
427C23#53911yescharacteristic
428C23#54211yescharacteristic
429C23#56211yescharacteristic
430C23#56211yescharacteristic
431C23#56311yescharacteristic
432C23#56311yescharacteristic
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—161no—
463C2×C4—161no—
464C2×C4—161no—
465C2×C4C4×#4111yescharacteristic
466C2×C4#51811yescharacteristic
467C2×C4#51811yescharacteristic
468C2×C4#51811yescharacteristic
469C2×C4#52011yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#53811yescharacteristic
472C2×C4#53811yescharacteristic
473C2×C4#53811yescharacteristic
474C2×C4#53911yescharacteristic
475C2×C4#53911yescharacteristic
476C2×C4#54011yescharacteristic
477C2×C4#54011yescharacteristic
478C2×C4#54111yescharacteristic
479C2×C4#54211yescharacteristic
480C2×C4#54211yescharacteristic
481C2×C4#56011yescharacteristic
482C2×C4#56011yescharacteristic
483C2×C4#56011yescharacteristic
484C2×C4#56011yescharacteristic
485C2×C4#56211yescharacteristic
486C2×C4#56211yescharacteristic
487C2×C4#56211yescharacteristic
488C2×C4#56211yescharacteristic
489C2×C4#56211yescharacteristic
490C2×C4#56211yescharacteristic
491C2×C4#56311yescharacteristic
492C2×C4#56311yescharacteristic
493C8—42no—
494C8—42no—
495C8—42no—
496C8—42no—
497C8—42no—
498C8—42no—
499C8—42no—
500C8—81no—
501C8—81no—
502C8—81no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—81no—
511C22—161no—
512C22—161no—
513C22—161no—
514C22#233211yescharacteristic
515C22#237811yescharacteristic
516C22#237811yescharacteristic
517C22#282711yescharacteristic
518C22#282911yescharacteristic
519C22#283111yescharacteristic
520C22#283711yescharacteristic
521C4—82no—
522C4—82no—
523C4—81no—
524C4—161no—
525C4—161no—
526C4#233311yescharacteristic
527C4#233411yescharacteristic
528C4#237711yescharacteristic
529C4#237711yescharacteristic
530C4#282611yescharacteristic
531C4#283011yescharacteristic
532C4#283611yescharacteristic
533C4#283711yescharacteristic
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—161no—
542C2#1465311yescharacteristic
subgroupG/H
centerC4#2826
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14653
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11375C2semidirect
3. #11367C2semidirect
4. #11363C2semidirect
5. #11294C2semidirect
6. #11254C2semidirect
7. #11251C2semidirect
14. #10906C2semidirect
15. #10901C2semidirect
29. #2070C4semidirect
30. #2069C4semidirect

Maximal subgroups

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

subgroupindex
#236222
#113752
#113672
#113632
#112942
#112542
#112512
#111652
#111592
#111462
#111412
#109552
#109062
#109012