Finite Group Databasebeta

← #370285

Subgroups — #370285

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2#21175C212yesmaximal
3C2×#1945C211yesmaximal, characteristic
4#21156C211yesmaximal, characteristic
5#20455C211yesmaximal, characteristic
6C2×#1772C211yesmaximal, characteristic
7#11276C212yesmaximal
8#11242C212yesmaximal
9#10899C212yesmaximal
10#10870C211yesmaximal, characteristic
11#10862C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#424C2212yes—
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2317C2212yes—
24#2274C2212yes—
25#1952—24no—
26#1949—22no—
27#1949—22no—
28#1948—22no—
29#1947—22no—
30#1945C418yescomplemented
31#1944—22no—
32#1942—21no—
33#1942—21no—
34#1776—22no—
35C4×#107—21no—
36C4×#107—21no—
37#1772—24no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#340C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2212yes—
47C4⋊#95C2212yes—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2312yes—
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2311yescharacteristic
59C2×#108—22no—
60C2×#108—22no—
61C2×#108—22no—
62C2×#108—22no—
63C2×#108—22no—
64C2×#108C2×C414yes—
65#428—24no—
66#428—24no—
67#427—22no—
68#427—22no—
69#427C2×C414yes—
70#427Q814yes—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—22no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107D411yescharacteristic
80C2×#107D411yescharacteristic
81C2×#107D411yescharacteristic
82C2×#107D411yescharacteristic
83C2×#107Q814yes—
84C2×#106—22no—
85C2×#106D411yescharacteristic
86C2×#106D411yescharacteristic
87C2×#106D412yes—
88#424—24no—
89#424C2×C418yes—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#405—22no—
93#405—22no—
94#405—22no—
95#405D414yes—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8—22no—
99C2×C4×C8—21no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102C2×C4×C8C2×C414yes—
103#389—22no—
104#389—22no—
105#385—42no—
106#384—22no—
107#382—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—42no—
113C2×#95—42no—
114#343—24no—
115#343—24no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#339—22no—
125#339—22no—
126#339C2×C418yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2411yescharacteristic
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16C22×C414yes—
147C2×M16C2×D412yes—
148C2×M16#4112yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8C22×C412yes—
157C22×C8C22×C412yes—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D412yes—
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8#4112yes—
166C22×C8#4112yes—
167C22×C8#4112yes—
168#117—21no—
169#117—22no—
170#117—21no—
171#117—22no—
172#117—21no—
173#117—21no—
174#117—21no—
175#117—21no—
176#117C22×C414yes—
177#117C2×D412yes—
178#117C2×Q814yes—
179#117#4112yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×Q814yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22×C412yes—
214#108—28no—
215#108—28no—
216#107—24no—
217#107—24no—
218#107—22no—
219#107—22no—
220#107—42no—
221#107—42no—
222#107—44no—
223#107C4⋊C414yes—
224#107C4⋊C414yes—
225#106—22no—
226#106—22no—
227#106—44no—
228#106C4⋊C418yes—
229C4⋊C8—28no—
230C4⋊C8—28no—
231#97—24no—
232#97—42no—
233C4×C8—28no—
234C4×C8—24no—
235C4×C8—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—84no—
241C24H(5,𝔽2)11yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C42−1+411yescharacteristic
277C22×C42−1+411yescharacteristic
278C22×C42−1+411yescharacteristic
279M16—24no—
280M16—24no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—22no—
285M16—42no—
286M16—42no—
287M16—41no—
288M16—41no—
289M16—41no—
290M16—41no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8C2×C4⋊C412yes—
309C2×C8C2×C4⋊C412yes—
310C2×C8C4×D412yes—
311C2×C8C4×D412yes—
312C2×C8C4×D414yes—
313C2×C8C4×Q814yes—
314C2×C8#12112yes—
315C2×C8#12112yes—
316C2×C8C22⋊Q812yes—
317C2×C8C22⋊Q812yes—
318C2×C8#12514yes—
319C2×C8C4⋊Q814yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—24no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—162no—
337C4⋊C4—162no—
338C4⋊C4C2×C4⋊C414yes—
339C4⋊C4C2×C4⋊C414yes—
340C4⋊C4#12112yes—
341C4⋊C4#12112yes—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—24no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—24no—
349C42—22no—
350C42—22no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—162no—
359C42—162no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23#52011yescharacteristic
365C23#53711yescharacteristic
366C23#54211yescharacteristic
367C23D4211yescharacteristic
368C23#55311yescharacteristic
369C23#55411yescharacteristic
370C23#55411yescharacteristic
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×Q811yescharacteristic
410C2×C4C4×#4111yescharacteristic
411C2×C4#51911yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4#52511yescharacteristic
416C2×C4C2×#12511yescharacteristic
417C2×C4#52911yescharacteristic
418C2×C4C2×C4⋊Q811yescharacteristic
419C2×C4#53211yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#54111yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4#54811yescharacteristic
425C2×C4#54811yescharacteristic
426C2×C4D4×Q811yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4#55211yescharacteristic
429C2×C4#55611yescharacteristic
430C2×C4#55711yescharacteristic
431C2×C4#55711yescharacteristic
432C2×C4#55711yescharacteristic
433C2×C4#55711yescharacteristic
434C8—22no—
435C8—22no—
436C8—42no—
437C8—42no—
438C8—44no—
439C8#38914yes—
440C8#38914yes—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22#232311yescharacteristic
454C22D4×C4⋊C411yescharacteristic
455C22#237011yescharacteristic
456C22#241611yescharacteristic
457C22#245211yescharacteristic
458C22#250911yescharacteristic
459C22#258411yescharacteristic
460C22#329011yescharacteristic
461C22#329011yescharacteristic
462C22#329211yescharacteristic
463C22#329311yescharacteristic
464C4—44no—
465C4—44no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—84no—
470C4—81no—
471C4—81no—
472C4—162no—
473C4C2×#38911yescharacteristic
474C4C4×#12111yescharacteristic
475C4#235111yescharacteristic
476C4#236511yescharacteristic
477C4#241511yescharacteristic
478C4#245311yescharacteristic
479C4#250611yescharacteristic
480C4#254611yescharacteristic
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2#1404611yescharacteristic
486C2#2269011yescharacteristic
487C2#2269211yescharacteristic
488C2#2479711yescharacteristic
489C2#2479811yescharacteristic
490C2#2650111yescharacteristic
491C2#2650211yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1945C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#211752
C2×#19452
#211562
#204552
C2×#17722
#112762
#112422
#108992
#108702
#108622