Finite Group Databasebeta

← #370259

Subgroups — #370259

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C212yesmaximal
2C2×#1950C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C4×#428C211yesmaximal, characteristic
6#20431C211yesmaximal, characteristic
7#11270C212yesmaximal
8#11249C212yesmaximal
9#10896C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10847C211yesmaximal, characteristic
12C22×#108C2212yes—
13C2×#428C2212yes—
14C2×#428C2211yescharacteristic
15C2×#428C2211yescharacteristic
16C22×#107C2212yes—
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405C2212yes—
21C2×C4×M16C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1952—28no—
26#1950C418yescomplemented
27#1948—21no—
28#1948—21no—
29#1947—21no—
30#1947—21no—
31#1947—22no—
32#1947—21no—
33#1947—21no—
34#1776—21no—
35#1776—21no—
36C4×#107—21no—
37C4×#106—21no—
38#1768—22no—
39#1767—21no—
40C4×C4⋊C8—21no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#341C2212yes—
46#1736C2211yescharacteristic
47#1736C2212yes—
48#1736C2211yescharacteristic
49C2×#340C2212yes—
50#1729C2212yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C2×#108—24no—
66C2×#108—24no—
67C2×#108C2×C418yes—
68#428—22no—
69#428—22no—
70#428—22no—
71#428—24no—
72#428—24no—
73#428D414yes—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—24no—
79C2×#107—22no—
80C2×#107—42no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—22no—
84#423—22no—
85#423—22no—
86#423C2×C414yes—
87#423Q814yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D412yes—
96C2×C4⋊C8Q814yes—
97#405—24no—
98#405—42no—
99C4×M16—22no—
100C4×M16C2×C414yes—
101C2×#97—21no—
102C2×#97—22no—
103C2×C4×C8—21no—
104C2×C4×C8—22no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C2×#95—21no—
121C43—21no—
122#343—24no—
123#343—24no—
124#343—24no—
125#343—42no—
126#341C2×C418yes—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#336—22no—
136#336—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4C2411yescharacteristic
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16C22×C412yes—
162C2×M16C22×C414yes—
163C2×M16C22×C412yes—
164C2×M16C2×D412yes—
165C2×M16C2×Q814yes—
166C2×M16#4112yes—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×Q814yes—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117—21no—
184#117—21no—
185#117C2×D412yes—
186#117#4112yes—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4#4112yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42C2×Q812yes—
233C2×C42#4112yes—
234#108—216no—
235#107—24no—
236#107—24no—
237#107—44no—
238#107—44no—
239#107—44no—
240#106—24no—
241#106—24no—
242#106—44no—
243C4⋊C8—24no—
244C4⋊C8—24no—
245C4⋊C8—24no—
246C4⋊C8—24no—
247C4⋊C8C4⋊C414yes—
248C4⋊C8C4⋊C414yes—
249C4⋊C8C4⋊C418yes—
250#97—24no—
251#97—44no—
252C4×C8—24no—
253C4×C8—44no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260C24C2×#4111yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C4×Q812yes—
285C22×C4C4×Q812yes—
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12512yes—
291C22×C4#12512yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+411yescharacteristic
304C22×C42−1+411yescharacteristic
305C22×C42−1+411yescharacteristic
306M16—22no—
307M16—22no—
308M16—22no—
309M16—24no—
310M16—24no—
311M16—22no—
312M16—22no—
313M16—24no—
314M16—24no—
315M16—42no—
316M16—42no—
317M16—42no—
318M16—41no—
319M16—41no—
320M16—42no—
321M16C4×D414yes—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8C2×C4⋊C414yes—
343C2×C8C2×C4⋊C414yes—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4#12112yes—
367C4⋊C4#12112yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C42—22no—
373C42—22no—
374C42—41no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—81no—
379C42—81no—
380C42—82no—
381C42—82no—
382C42C2×C4⋊C412yes—
383C42C2×C4⋊C412yes—
384C42C4×D412yes—
385C42C4×D412yes—
386C42C22⋊Q812yes—
387C42C22⋊Q812yes—
388C42C4⋊Q812yes—
389C42C4⋊Q812yes—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23C2×C4×Q811yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×#12511yescharacteristic
397C23#54711yescharacteristic
398C23#55211yescharacteristic
399C23#55311yescharacteristic
400C23#55611yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—162no—
437C2×C4—162no—
438C2×C4#38912yes—
439C2×C4#38912yes—
440C2×C4#38912yes—
441C2×C4#38912yes—
442C2×C4#47212yes—
443C2×C4#47312yes—
444C2×C4C22×C4⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C4×#4111yescharacteristic
448C2×C4#51911yescharacteristic
449C2×C4#52011yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4#52511yescharacteristic
453C2×C4#52911yescharacteristic
454C2×C4C2×C4⋊Q811yescharacteristic
455C2×C4#53211yescharacteristic
456C2×C4#53611yescharacteristic
457C2×C4#53711yescharacteristic
458C2×C4#54111yescharacteristic
459C2×C4#54211yescharacteristic
460C2×C4D4211yescharacteristic
461C2×C4#54711yescharacteristic
462C2×C4#54811yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4D4×Q811yescharacteristic
465C2×C4D4×Q811yescharacteristic
466C2×C4#55411yescharacteristic
467C2×C4#55411yescharacteristic
468C2×C4#55711yescharacteristic
469C2×C4#55711yescharacteristic
470C2×C4#55711yescharacteristic
471C2×C4#55711yescharacteristic
472C8—24no—
473C8—24no—
474C8—24no—
475C8—42no—
476C8—44no—
477C8—44no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22C2×#38911yescharacteristic
491C22#236511yescharacteristic
492C22#237011yescharacteristic
493C22#241511yescharacteristic
494C22#241611yescharacteristic
495C22#250611yescharacteristic
496C22#250911yescharacteristic
497C22C2×#47211yescharacteristic
498C22C2×#47311yescharacteristic
499C22#306811yescharacteristic
500C22#306911yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—82no—
508C4—81no—
509C4—82no—
510C4—162no—
511C4#195214yescomplemented
512C4C4×#12111yescharacteristic
513C4#232311yescharacteristic
514C4D4×C4⋊C411yescharacteristic
515C4#235111yescharacteristic
516C4#245211yescharacteristic
517C4#245311yescharacteristic
518C4#254611yescharacteristic
519C4#258411yescharacteristic
520C2—21no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2#1404611yescharacteristic
525C2C2×#195211yescharacteristic
526C2#2118211yescharacteristic
527C2#2472011yescharacteristic
528C2#2472211yescharacteristic
529C2#2477011yescharacteristic
530C2#2477211yescharacteristic
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
26. #1950C4semidirect
511. C4#1952semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
C2×#19502
#211712
C2×#19472
C4×#4282
#204312
#112702
#112492
#108962
#108572
#108472