Finite Group Databasebeta

← #367568

Subgroups — #367568

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2C2×#1952C211yesmaximal, characteristic
3#21160C212yesmaximal
4#21158C211yesmaximal, characteristic
5#20803C211yesmaximal, characteristic
6C2×#1772C211yesmaximal, characteristic
7#11342C212yesmaximal
8#11217C212yesmaximal
9#10896C212yesmaximal
10#10862C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#427C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2212yes—
21C2×#405C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2317C2212yes—
24#2316C2212yes—
25#1952—24no—
26#1952C418yescomplemented
27#1944—22no—
28#1943—22no—
29#1943—22no—
30#1942—21no—
31#1942—21no—
32#1942—22no—
33#1941—22no—
34#1847—22no—
35#1846—21no—
36#1845—21no—
37#1772—24no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2211yescharacteristic
41#1736C2212yes—
42#1735C2211yescharacteristic
43C2×#340C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C4⋊#95C2212yes—
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2312yes—
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42C2311yescharacteristic
59C2×#108C2×C414yes—
60C2×#108D414yes—
61C2×#108Q814yes—
62#428—24no—
63#428—24no—
64#428—24no—
65#428—42no—
66#428—42no—
67#428C2×C418yes—
68#427—22no—
69#427—22no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107C2×C414yes—
81C2×#107D411yescharacteristic
82C2×#107D411yescharacteristic
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106D412yes—
87#405—22no—
88#405—24no—
89#405—22no—
90#405—22no—
91#405—42no—
92#405C2×C414yes—
93C2×#97—21no—
94C2×#97—21no—
95C2×#97—22no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8Q814yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#385—42no—
109#384—22no—
110#382—22no—
111#380—42no—
112C2×#95—42no—
113#343—24no—
114#343C2×C418yes—
115#341Q814yes—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340D411yescharacteristic
121#340D411yescharacteristic
122#340D411yescharacteristic
123#340D411yescharacteristic
124#339—24no—
125#339—22no—
126#339—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2411yescharacteristic
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16C22×C414yes—
143C2×M16C22×C412yes—
144C2×M16C2×D414yes—
145C2×M16C2×Q812yes—
146C2×M16C2×Q814yes—
147C2×M16C2×Q812yes—
148C2×M16#4112yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C22×C412yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D412yes—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×Q812yes—
163C22×C8C2×Q812yes—
164C22×C8C2×Q812yes—
165C22×C8C2×Q812yes—
166C22×C8#4112yes—
167C22×C8#4112yes—
168#117—21no—
169#117—21no—
170#117—22no—
171#117—21no—
172#117—21no—
173#117—21no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—21no—
178#117C22×C412yes—
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⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42C2×Q812yes—
214#108C4⋊C4116yescomplemented
215#107—24no—
216#107—24no—
217#107—22no—
218#107—22no—
219#107—28no—
220#107—44no—
221#107—44no—
222#107—42no—
223#107—42no—
224#106—24no—
225#106—28no—
226#106—44no—
227#106—44no—
228#97—24no—
229#97—44no—
230#97—42no—
231C4×C8—24no—
232C4×C8—28no—
233C4×C8—44no—
234C4×C8—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—84no—
240C24H(5,𝔽2)11yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C42−1+411yescharacteristic
277C22×C42−1+411yescharacteristic
278M16—24no—
279M16—22no—
280M16—24no—
281M16—22no—
282M16—24no—
283M16—22no—
284M16—22no—
285M16—22no—
286M16—41no—
287M16—41no—
288M16C2×C4⋊C418yes—
289M16C4×Q814yes—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—24no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8C2×C4⋊C414yes—
308C2×C8C4×D412yes—
309C2×C8C4×D412yes—
310C2×C8C4×Q814yes—
311C2×C8#12112yes—
312C2×C8#12112yes—
313C2×C8C22⋊Q814yes—
314C2×C8#12514yes—
315C2×C8C4⋊Q812yes—
316C2×C8C4⋊Q812yes—
317C2×C8C4⋊Q812yes—
318C2×C8C4⋊Q812yes—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—162no—
338C4⋊C4—162no—
339C4⋊C4—162no—
340C4⋊C4—162no—
341C4⋊C4C4×D412yes—
342C4⋊C4C4×D412yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—42no—
347C22⋊C4—41no—
348C22⋊C4—42no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—41no—
357C42—41no—
358C42—41no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23#52011yescharacteristic
364C23#54011yescharacteristic
365C23#54411yescharacteristic
366C23D4211yescharacteristic
367C23#55011yescharacteristic
368C23#55311yescharacteristic
369C23Q8211yescharacteristic
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—162no—
404C2×C4—162no—
405C2×C4C22×C4⋊C411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×Q811yescharacteristic
409C2×C4C2×C4×Q811yescharacteristic
410C2×C4#52011yescharacteristic
411C2×C4C2×#12111yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4C2×#12511yescharacteristic
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4C2×C4⋊Q811yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4#53311yescharacteristic
418C2×C4#53311yescharacteristic
419C2×C4#54111yescharacteristic
420C2×C4#54411yescharacteristic
421C2×C4D4211yescharacteristic
422C2×C4#54611yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#54811yescharacteristic
426C2×C4D4×Q811yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4D4×Q811yescharacteristic
429C2×C4#55211yescharacteristic
430C2×C4#55411yescharacteristic
431C2×C4#55411yescharacteristic
432C2×C4#55711yescharacteristic
433C8—28no—
434C8—24no—
435C8—22no—
436C8—22no—
437C8—42no—
438C8#38914yes—
439C8#38914yes—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22#233011yescharacteristic
453C22Q8×C4⋊C411yescharacteristic
454C22#236111yescharacteristic
455C22#242811yescharacteristic
456C22#244111yescharacteristic
457C22#250711yescharacteristic
458C22#255111yescharacteristic
459C22#329011yescharacteristic
460C22#329011yescharacteristic
461C22#329211yescharacteristic
462C22#329311yescharacteristic
463C4—44no—
464C4—42no—
465C4—42no—
466C4—84no—
467C4—82no—
468C4—81no—
469C4—81no—
470C4—82no—
471C4—162no—
472C4C2×#38911yescharacteristic
473C4C4×C4⋊Q811yescharacteristic
474C4D4×C4⋊C411yescharacteristic
475C4#236011yescharacteristic
476C4#242911yescharacteristic
477C4#244011yescharacteristic
478C4#250611yescharacteristic
479C4#255011yescharacteristic
480C2—21no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2#1400511yescharacteristic
485C2#2304511yescharacteristic
486C2#2304611yescharacteristic
487C2#2544111yescharacteristic
488C2#2544211yescharacteristic
489C2#2882311yescharacteristic
490C2#2882511yescharacteristic
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
26. #1952C4semidirect
214. #108C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#211602
#211582
#208032
C2×#17722
#113422
#112172
#108962
#108622
#107062