Finite Group Databasebeta

← #370284

Subgroups — #370284

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1951C211yesmaximal, characteristic
2#21175C212yesmaximal
3C2×#1945C211yesmaximal, characteristic
4#21155C211yesmaximal, characteristic
5#20454C211yesmaximal, characteristic
6C2×#1772C211yesmaximal, characteristic
7#11276C212yesmaximal
8#11242C212yesmaximal
9#10899C212yesmaximal
10#10869C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#424C2212yes—
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2317C2212yes—
24#2274C2212yes—
25#1951—24no—
26#1949—22no—
27#1949—22no—
28#1948—22no—
29#1947—22no—
30#1945C418yescomplemented
31#1943—21no—
32#1943—21no—
33#1941—22no—
34#1776—21no—
35#1776—21no—
36C4×#106—22no—
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—22no—
72C2×#107D412yes—
73C2×#107D411yescharacteristic
74C2×#107D411yescharacteristic
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—22no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106D412yes—
84C2×#106D412yes—
85C2×#106Q814yes—
86#424—24no—
87#424C2×C418yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90#405—22no—
91#405—22no—
92#405—22no—
93#405D414yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8—22no—
99C2×C4×C8—21no—
100C2×C4×C8C2×C414yes—
101#389—22no—
102#389—22no—
103#385—42no—
104#384—22no—
105#382—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C22⋊C4—42no—
111C2×#95—42no—
112#343—24no—
113#343—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#339—22no—
123#339—22no—
124#339C2×C418yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2411yescharacteristic
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16C22×C414yes—
145C2×M16C2×D412yes—
146C2×M16#4112yes—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D412yes—
161C22×C8C2×Q812yes—
162C22×C8C2×Q812yes—
163C22×C8#4112yes—
164C22×C8#4112yes—
165C22×C8#4112yes—
166#117—21no—
167#117—22no—
168#117—21no—
169#117—22no—
170#117—21no—
171#117—21no—
172#117—21no—
173#117—21no—
174#117C22×C414yes—
175#117C2×D412yes—
176#117C2×Q814yes—
177#117#4112yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22×C412yes—
210#108—28no—
211#108—28no—
212#107—24no—
213#107—44no—
214#107C4⋊C414yes—
215#107C4⋊C414yes—
216#106—24no—
217#106—24no—
218#106—24no—
219#106—42no—
220#106—42no—
221#106—44no—
222#106C4⋊C418yes—
223C4⋊C8—28no—
224C4⋊C8—28no—
225#97—24no—
226#97—42no—
227C4×C8—28no—
228C4×C8—24no—
229C4×C8—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—84no—
235C24H(5,𝔽2)11yescharacteristic
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×Q811yescharacteristic
260C22×C4C22×Q811yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C42−1+411yescharacteristic
267C22×C42−1+411yescharacteristic
268C22×C42−1+411yescharacteristic
269M16—24no—
270M16—24no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—22no—
275M16—42no—
276M16—42no—
277M16—41no—
278M16—41no—
279M16—41no—
280M16—41no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8C2×C4⋊C412yes—
299C2×C8C2×C4⋊C412yes—
300C2×C8C4×D412yes—
301C2×C8C4×D412yes—
302C2×C8C4×D414yes—
303C2×C8C4×Q814yes—
304C2×C8#12112yes—
305C2×C8#12112yes—
306C2×C8C22⋊Q812yes—
307C2×C8C22⋊Q812yes—
308C2×C8#12514yes—
309C2×C8C4⋊Q814yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—162no—
324C4⋊C4—162no—
325C4⋊C4C2×C4⋊C414yes—
326C4⋊C4C2×C4⋊C414yes—
327C4⋊C4#12114yes—
328C22⋊C4—24no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—24no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—162no—
345C42—162no—
346C23—22no—
347C23—22no—
348C23#52011yescharacteristic
349C23#53711yescharacteristic
350C23#54211yescharacteristic
351C23D4211yescharacteristic
352C23#55311yescharacteristic
353C23#55411yescharacteristic
354C23#55411yescharacteristic
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—162no—
381C2×C4—162no—
382C2×C4C22×C4⋊C411yescharacteristic
383C2×C4C2×C4×D411yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×Q811yescharacteristic
386C2×C4C4×#4111yescharacteristic
387C2×C4#51911yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4#52511yescharacteristic
392C2×C4C2×#12511yescharacteristic
393C2×C4#52911yescharacteristic
394C2×C4C2×C4⋊Q811yescharacteristic
395C2×C4#53211yescharacteristic
396C2×C4#53611yescharacteristic
397C2×C4#54111yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4#54711yescharacteristic
400C2×C4#54811yescharacteristic
401C2×C4#54811yescharacteristic
402C2×C4D4×Q811yescharacteristic
403C2×C4D4×Q811yescharacteristic
404C2×C4#55211yescharacteristic
405C2×C4#55611yescharacteristic
406C2×C4#55711yescharacteristic
407C2×C4#55711yescharacteristic
408C2×C4#55711yescharacteristic
409C2×C4#55711yescharacteristic
410C8—22no—
411C8—22no—
412C8—42no—
413C8—42no—
414C8—44no—
415C8#38914yes—
416C8#38914yes—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—22no—
425C22#232311yescharacteristic
426C22D4×C4⋊C411yescharacteristic
427C22#237011yescharacteristic
428C22#241611yescharacteristic
429C22#245211yescharacteristic
430C22#250911yescharacteristic
431C22#258411yescharacteristic
432C22#329012yes—
433C22#329112yes—
434C4—44no—
435C4—44no—
436C4—44no—
437C4—44no—
438C4—84no—
439C4—82no—
440C4—162no—
441C4C2×#38911yescharacteristic
442C4C4×#12111yescharacteristic
443C4#235111yescharacteristic
444C4#236511yescharacteristic
445C4#241511yescharacteristic
446C4#245311yescharacteristic
447C4#250611yescharacteristic
448C4#254611yescharacteristic
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2#1404611yescharacteristic
454C2#2268911yescharacteristic
455C2#2269111yescharacteristic
456C2#2479612yes—
457C2#2650012yes—
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×#19512
#211752
C2×#19452
#211552
#204542
C2×#17722
#112762
#112422
#108992
#108692
#108612