Finite Group Databasebeta

← #360412

Subgroups — #360412

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20765C211yesmaximal, characteristic
2#20647C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic
5C2×C4×#101C211yesmaximal, characteristic
6#11303C212yesmaximal
7#11164C212yesmaximal
8#10942C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10853C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#419C2212yes—
16C22×#103C2212yes—
17C22×#101C2211yescharacteristic
18C22×#101C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24#1836—22no—
25#1835—22no—
26#1805—21no—
27#1805—21no—
28#1805—22no—
29#1805—22no—
30#1803—22no—
31#1771—22no—
32#1770—21no—
33C8×C4⋊C4—21no—
34C4×#101C418yescomplemented
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2212yes—
46C23×Q8C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C23×C8C2312yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C414yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#423—22no—
68#423—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D412yes—
76#419—24no—
77#419—42no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#101—22no—
84C2×#101—22no—
85C2×#101—22no—
86C2×#101—22no—
87C2×#101C2×C414yes—
88C2×#101C2×C414yes—
89#405—24no—
90#405—42no—
91C4×M16—22no—
92C4×M16—22no—
93C4×M16C2×C414yes—
94C4×M16C2×C414yes—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—22no—
101#384D412yes—
102#382—21no—
103#382—21no—
104#382—21no—
105#382—21no—
106#382D412yes—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C2×#95—21no—
112C2×#95—21no—
113#343—24no—
114#343—42no—
115#343C2×C414yes—
116#343C2×C414yes—
117#341—42no—
118#341—42no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—22no—
125#336—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×Q8#4112yes—
139C22×Q8#4112yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4C2411yescharacteristic
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
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—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16C22×C412yes—
171C2×M16C22×C412yes—
172C2×M16C22×C412yes—
173C2×M16C22×C412yes—
174C2×M16#4112yes—
175C2×M16#4112yes—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—22no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239#103—48no—
240#103—44no—
241#103—44no—
242#101—28no—
243#101—28no—
244#97—24no—
245#97—24no—
246#97—44no—
247C4×C8—24no—
248C4×C8—24no—
249C4×C8—44no—
250#95—22no—
251#95—22no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24C2×#4111yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8#11714yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4#11712yes—
305C22×C4#11712yes—
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12612yes—
311C22×C4#12612yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326M16—22no—
327M16—22no—
328M16—22no—
329M16—22no—
330M16—22no—
331M16—22no—
332M16—22no—
333M16—22no—
334M16—22no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—42no—
342M16—42no—
343M16—42no—
344M16—42no—
345M16#11714yes—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—82no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—42no—
388C42—42no—
389C42—44no—
390C42—44no—
391C42C4×D412yes—
392C42C4×D412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12312yes—
396C42#12312yes—
397C42#12412yes—
398C42#12412yes—
399C23—22no—
400C23—22no—
401C23C2×#11711yescharacteristic
402C23C2×#12111yescharacteristic
403C23C2×#12611yescharacteristic
404C23#52911yescharacteristic
405C23#52911yescharacteristic
406C23#54211yescharacteristic
407C23#54211yescharacteristic
408Q8—24no—
409Q8—24no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—162no—
442C2×C4—162no—
443C2×C4#38812yes—
444C2×C4#38812yes—
445C2×C4#38812yes—
446C2×C4#38812yes—
447C2×C4C2×#11711yescharacteristic
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×C4×D411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4#52011yescharacteristic
452C2×C4#52011yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×#12411yescharacteristic
455C2×C4#52911yescharacteristic
456C2×C4#52911yescharacteristic
457C2×C4#54011yescharacteristic
458C2×C4#54111yescharacteristic
459C2×C4#54611yescharacteristic
460C2×C4#54611yescharacteristic
461C2×C4#54611yescharacteristic
462C2×C4#54711yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4#54811yescharacteristic
465C2×C4#54811yescharacteristic
466C2×C4#55011yescharacteristic
467C2×C4#55211yescharacteristic
468C2×C4#55211yescharacteristic
469C2×C4#55211yescharacteristic
470C2×C4#55311yescharacteristic
471C2×C4#55311yescharacteristic
472C2×C4#55311yescharacteristic
473C2×C4#55511yescharacteristic
474C2×C4#55611yescharacteristic
475C8—24no—
476C8—42no—
477C8—42no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—82no—
482C22—22no—
483C22—22no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22#228311yescharacteristic
491C22C2×#38811yescharacteristic
492C22#231411yescharacteristic
493C22#242611yescharacteristic
494C22#242711yescharacteristic
495C22#249411yescharacteristic
496C22#249511yescharacteristic
497C22#306712yes—
498C22#306912yes—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—84no—
505C4—82no—
506C4—162no—
507C4#235211yescharacteristic
508C4#235211yescharacteristic
509C4#235211yescharacteristic
510C4#235411yescharacteristic
511C4#242011yescharacteristic
512C4#242111yescharacteristic
513C4#248911yescharacteristic
514C4#249011yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—21no—
519C2#1387511yescharacteristic
520C2#2107511yescharacteristic
521C2#2107711yescharacteristic
522C2#2531712yes—
523C2#2536312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#207652
#206472
C2×#18052
#204412
C2×C4×#1012
#113032
#111642
#109422
#108582
#108532
#108272