Finite Group Databasebeta

← #365864

Subgroups — #365864

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2#21035C212yesmaximal
3#20803C211yesmaximal, characteristic
4#20455C211yesmaximal, characteristic
5#11290C212yesmaximal
6#11217C212yesmaximal
7#10899C212yesmaximal
8#10870C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2317C2212yes—
22#2274C2212yes—
23#1952C418yescomplemented
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28#1847—22no—
29#1846—21no—
30#1845—21no—
31#1776—22no—
32C4×#107—21no—
33C4×#107—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37#1736C2212yes—
38#1736C2212yes—
39#1735C2211yescharacteristic
40C2×#340C2212yes—
41C2×#339C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2211yescharacteristic
44#1729C2212yes—
45C4⋊#95C2212yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C2×#108—22no—
58C2×#108—22no—
59C2×#108C2×C414yes—
60#428—24no—
61#428—24no—
62#428—42no—
63#428C2×C418yes—
64#427—22no—
65#427—22no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107C2×C414yes—
76C2×#107D411yescharacteristic
77C2×#107D411yescharacteristic
78C2×#106—22no—
79C2×#106—22no—
80C2×#106D412yes—
81#405—22no—
82#405—22no—
83#405C2×C414yes—
84#405Q814yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#389—22no—
93#389—22no—
94#385—42no—
95#384—22no—
96#382—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—42no—
102C2×#95—42no—
103#343—42no—
104#343C2×C418yes—
105#341—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340D411yescharacteristic
113#340D411yescharacteristic
114#340D411yescharacteristic
115#340D411yescharacteristic
116#339—22no—
117#339—42no—
118#339Q814yes—
119#336—22no—
120#336—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C2411yescharacteristic
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16C22×C414yes—
139C2×M16C22×C412yes—
140C2×M16C2×Q812yes—
141C2×M16C2×Q812yes—
142C2×M16#4112yes—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C22×C412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162#117—21no—
163#117—21no—
164#117—21no—
165#117—21no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117C22×C412yes—
171#117C2×Q812yes—
172#117C2×Q812yes—
173#117#4112yes—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
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—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×Q812yes—
208#108—28no—
209#107—24no—
210#107—24no—
211#107—22no—
212#107—22no—
213#107—28no—
214#107—44no—
215#107—42no—
216#107—42no—
217#106—24no—
218#106—28no—
219#106—44no—
220#97—24no—
221#97—24no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—44no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—84no—
230C24H(5,𝔽2)11yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C42−1+411yescharacteristic
267C22×C42−1+411yescharacteristic
268M16—22no—
269M16—22no—
270M16—22no—
271M16—24no—
272M16—24no—
273M16—22no—
274M16—22no—
275M16—42no—
276M16—41no—
277M16—42no—
278M16—41no—
279M16C4×Q814yes—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8#11714yes—
300C2×C8C4×D412yes—
301C2×C8C4×D412yes—
302C2×C8C4×Q814yes—
303C2×C8C22⋊Q812yes—
304C2×C8C22⋊Q812yes—
305C2×C8#12312yes—
306C2×C8#12312yes—
307C2×C8#12514yes—
308C2×C8#12614yes—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—162no—
326C4⋊C4—162no—
327C4⋊C4C4×D412yes—
328C4⋊C4C4×D412yes—
329C4⋊C4C4⋊Q812yes—
330C4⋊C4C4⋊Q812yes—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—162no—
348C42—162no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#52011yescharacteristic
354C23#54011yescharacteristic
355C23#54411yescharacteristic
356C23D4211yescharacteristic
357C23#55311yescharacteristic
358C23#55411yescharacteristic
359C23#55911yescharacteristic
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—162no—
394C2×C4—162no—
395C2×C4C2×#11711yescharacteristic
396C2×C4C2×C4×D411yescharacteristic
397C2×C4C2×C4×D411yescharacteristic
398C2×C4C2×C4×Q811yescharacteristic
399C2×C4C2×C4×Q811yescharacteristic
400C2×C4#52011yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4C2×#12311yescharacteristic
403C2×C4C2×#12511yescharacteristic
404C2×C4C2×#12611yescharacteristic
405C2×C4C2×C4⋊Q811yescharacteristic
406C2×C4#53211yescharacteristic
407C2×C4#53311yescharacteristic
408C2×C4#53311yescharacteristic
409C2×C4#54111yescharacteristic
410C2×C4#54411yescharacteristic
411C2×C4#54611yescharacteristic
412C2×C4#54711yescharacteristic
413C2×C4#54711yescharacteristic
414C2×C4#54811yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C2×C4#55011yescharacteristic
417C2×C4#55211yescharacteristic
418C2×C4#55411yescharacteristic
419C2×C4#55511yescharacteristic
420C2×C4#55611yescharacteristic
421C2×C4#55611yescharacteristic
422C2×C4#55711yescharacteristic
423C8—24no—
424C8—44no—
425C8—42no—
426C8—41no—
427C8—41no—
428C8#38714yes—
429C8#38714yes—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22#233011yescharacteristic
443C22#235511yescharacteristic
444C22#236111yescharacteristic
445C22#244011yescharacteristic
446C22#244511yescharacteristic
447C22#249711yescharacteristic
448C22#257011yescharacteristic
449C22#329011yescharacteristic
450C22#329011yescharacteristic
451C22#329211yescharacteristic
452C22#329311yescharacteristic
453C4—44no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4—162no—
462C4C2×#38711yescharacteristic
463C4C4×C4⋊Q811yescharacteristic
464C4#235411yescharacteristic
465C4#236011yescharacteristic
466C4#243911yescharacteristic
467C4#244611yescharacteristic
468C4#249611yescharacteristic
469C4#255211yescharacteristic
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2#1398111yescharacteristic
475C2#2304511yescharacteristic
476C2#2304611yescharacteristic
477C2#2608211yescharacteristic
478C2#2608311yescharacteristic
479C2#2627111yescharacteristic
480C2#2627211yescharacteristic
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
23. #1952C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#210352
#208032
#204552
#112902
#112172
#108992
#108702
#108622
#108592
#107062