Finite Group Databasebeta

← #367605

Subgroups — #367605

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1950C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#20441C211yesmaximal, characteristic
6#11270C212yesmaximal
7#11249C212yesmaximal
8#10896C212yesmaximal
9#10870C212yesmaximal
10#10857C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#107C2212yes—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#405C2212yes—
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1950C418yescomplemented
24#1949—22no—
25#1948—21no—
26#1947—21no—
27#1947—21no—
28#1947—22no—
29#1947—21no—
30#1836—22no—
31#1834—21no—
32#1834—21no—
33#1771—22no—
34#1770—21no—
35C8×C4⋊C4—21no—
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38C2×#343C2211yescharacteristic
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#343C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C23×C8C2312yes—
54C2×#117C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C2×#108—42no—
63C2×#108—42no—
64C2×#108C2×C418yes—
65#428—24no—
66#428—42no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—24no—
71C2×#107—42no—
72C2×#106—22no—
73#423—22no—
74#423C2×C414yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2×C414yes—
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8D412yes—
81C2×C4⋊C8Q814yes—
82#405—24no—
83#405—42no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16C2×C414yes—
87C4×M16Q814yes—
88C2×#97—21no—
89C2×#97—21no—
90C2×#97—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#382D412yes—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C2×#95—21no—
110#343—22no—
111#343—22no—
112#343—24no—
113#343—22no—
114#343—22no—
115#343—42no—
116#341C2×C418yes—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#336—22no—
126#336—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16C22×C412yes—
157C2×M16C22×C414yes—
158C2×M16C22×C412yes—
159C2×M16C2×Q812yes—
160C2×M16C2×Q812yes—
161C2×M16#4112yes—
162C2×M16#4112yes—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8C22×C414yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×Q814yes—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4#4112yes—
209C2×C4⋊C4#4112yes—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—82no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q812yes—
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223#108—48no—
224#107—44no—
225#107—44no—
226#107—44no—
227#106—44no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8C4⋊C414yes—
231C4⋊C8C4⋊C414yes—
232C4⋊C8C4⋊C418yes—
233#97—24no—
234#97—24no—
235#97—44no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—44no—
239#95—22no—
240#95—22no—
241#95—41no—
242#95—41no—
243C24C2×#4111yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C4×Q812yes—
266C22×C4C4×Q812yes—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12512yes—
272C22×C4#12512yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286C22×C42−1+411yescharacteristic
287M16—22no—
288M16—22no—
289M16—22no—
290M16—22no—
291M16—22no—
292M16—24no—
293M16—22no—
294M16—24no—
295M16—22no—
296M16—22no—
297M16—22no—
298M16—41no—
299M16—41no—
300M16—42no—
301M16—41no—
302M16—42no—
303M16—41no—
304M16—41no—
305M16—41no—
306M16#12514yes—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8C2×C4⋊C414yes—
331C2×C8C2×C4⋊C414yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C42—42no—
359C42—42no—
360C42—82no—
361C42—82no—
362C42C2×C4⋊C412yes—
363C42C2×C4⋊C412yes—
364C42C4×D412yes—
365C42C4×D412yes—
366C42C22⋊Q812yes—
367C42C22⋊Q812yes—
368C42C4⋊Q812yes—
369C42C4⋊Q812yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23C2×C4×Q811yescharacteristic
375C23C2×#12111yescharacteristic
376C23C2×#12511yescharacteristic
377C23#52911yescharacteristic
378C23#52911yescharacteristic
379C23#54211yescharacteristic
380C23#54211yescharacteristic
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—162no—
411C2×C4—162no—
412C2×C4#38912yes—
413C2×C4#38912yes—
414C2×C4#38912yes—
415C2×C4#38912yes—
416C2×C4#47212yes—
417C2×C4#47312yes—
418C2×C4C22×C4⋊C411yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C4×#4111yescharacteristic
421C2×C4C4×#4111yescharacteristic
422C2×C4#51811yescharacteristic
423C2×C4#51911yescharacteristic
424C2×C4C2×C22⋊Q811yescharacteristic
425C2×C4C2×#12511yescharacteristic
426C2×C4C2×C4⋊Q811yescharacteristic
427C2×C4#53311yescharacteristic
428C2×C4#54411yescharacteristic
429C2×C4#54411yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4#54611yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#54811yescharacteristic
435C2×C4#54811yescharacteristic
436C2×C4#54811yescharacteristic
437C2×C4#55011yescharacteristic
438C2×C4#55211yescharacteristic
439C2×C4#55311yescharacteristic
440C2×C4#55411yescharacteristic
441C2×C4#55411yescharacteristic
442C2×C4#55511yescharacteristic
443C2×C4#55611yescharacteristic
444C2×C4#55711yescharacteristic
445C2×C4#55711yescharacteristic
446C8—24no—
447C8—24no—
448C8—24no—
449C8—42no—
450C8—42no—
451C8—44no—
452C8—42no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22C2×#38911yescharacteristic
466C22C4×#12511yescharacteristic
467C22#233111yescharacteristic
468C22#242611yescharacteristic
469C22#242711yescharacteristic
470C22#250811yescharacteristic
471C22#250911yescharacteristic
472C22C2×#47211yescharacteristic
473C22C2×#47311yescharacteristic
474C22#306811yescharacteristic
475C22#306911yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#195214yes—
485C4#234911yescharacteristic
486C4#235111yescharacteristic
487C4#236211yescharacteristic
488C4#236311yescharacteristic
489C4#243911yescharacteristic
490C4#244211yescharacteristic
491C4#255011yescharacteristic
492C4#255211yescharacteristic
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2#1400611yescharacteristic
498C2C2×#195211yescharacteristic
499C2#2118211yescharacteristic
500C2#2531211yescharacteristic
501C2#2531411yescharacteristic
502C2#2535811yescharacteristic
503C2#2536011yescharacteristic
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
23. #1950C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19502
#211702
C2×#19472
#207642
#204412
#112702
#112492
#108962
#108702
#108572
C4×#3432