Finite Group Databasebeta

← #365679

Subgroups — #365679

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic
6#11296C214yesmaximal
7#10899C214yesmaximal
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2369—21no—
15#2369—21no—
16#2365—21no—
17#2365—21no—
18#2345—22no—
19Q8×C22⋊C4—22no—
20#2331—22no—
21#2316C2212yes—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C2×#391—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2274C2212yes—
28C4×#117C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1997C418yescomplemented
31#1948—21no—
32#1948—21no—
33#1947—21no—
34#1947—21no—
35#1836—22no—
36#1834—21no—
37#1834—21no—
38C2×#343C2214yes—
39#1736C2214yes—
40#1736C2212yes—
41C2×#339C2212yes—
42C2×#339C2214yes—
43C2×#339C2212yes—
44C4⋊#95—22no—
45C4⋊#95C2214yes—
46C23×Q8D412yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8Q814yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×#117C2311yescharacteristic
60C2×#117C2312yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#428—28no—
72C2×#107—22no—
73C2×#106—22no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C4×M16—22no—
79C4×M16C2×C414yes—
80C2×#97—21no—
81C2×C4×C8—21no—
82#391—22no—
83#391—22no—
84#391—24no—
85#391—24no—
86#391—48no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#387—24no—
94#387—24no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#382D412yes—
102#380—24no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—21no—
120#376—21no—
121C2×#95—28no—
122C43—21no—
123C43—22no—
124C43—22no—
125#343C2×C4116yes—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#339—24no—
131#339—24no—
132#339—28no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8C4⋊C418yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D414yes—
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—24no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16C22×C414yes—
154C2×M16C22×C414yes—
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4×Q8—84no—
166C4×Q8—84no—
167#117—22no—
168#117—24no—
169#117—22no—
170#117—24no—
171#117—22no—
172#117#4114yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C4⋊C418yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×Q814yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×Q812yes—
234C2×C42#4114yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#107—44no—
238#106—44no—
239C4⋊C8—24no—
240C4⋊C8—24no—
241#97—24no—
242C4×C8—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—416no—
254C24C22×D411yescharacteristic
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—416no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×Q8—88no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4C2×C4⋊C418yes—
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4112yes—
305C22×C42−1+412yes—
306C22×C42−1+412yes—
307M16—24no—
308M16—28no—
309M16—24no—
310M16—44no—
311M16—42no—
312M16—42no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4#12412yes—
347C4⋊C4#12412yes—
348C22⋊C4—24no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—48no—
355C42—48no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—84no—
366C42—84no—
367C42C4×D412yes—
368C42C4×D412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23C22×C4⋊C411yescharacteristic
380C23C2×#12011yescharacteristic
381C23C2×#12311yescharacteristic
382C23#54812yes—
383C23#54812yes—
384Q8—416no—
385Q8—88no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—48no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—84no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#38512yes—
429C2×C4#38512yes—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4#45312yes—
433C2×C4#45412yes—
434C2×C4#45512yes—
435C2×C4#45612yes—
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C4×#4112yes—
439C2×C4#51912yes—
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×#12411yescharacteristic
443C2×C4#52912yes—
444C2×C4#53211yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#54112yes—
448C2×C4#54712yes—
449C2×C4#54812yes—
450C2×C4#55212yes—
451C2×C4#55612yes—
452C2×C4#55712yes—
453C2×C4#55712yes—
454C8—44no—
455C8—44no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22C2×#38511yescharacteristic
465C22#235112yes—
466C22#242112yes—
467C22#243211yescharacteristic
468C22#243611yescharacteristic
469C22C2×#45311yescharacteristic
470C22C2×#45511yescharacteristic
471C22C2×#45411yescharacteristic
472C22C2×#45611yescharacteristic
473C4—22no—
474C4—22no—
475C4—48no—
476C4—42no—
477C4—42no—
478C4—48no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—82no—
483C4—162no—
484C4#185614yes—
485C4#185714yes—
486C4C4×#12411yescharacteristic
487C4#232711yescharacteristic
488C4#236212yes—
489C4#245312yes—
490C4#256511yescharacteristic
491C4#256811yescharacteristic
492C2—22no—
493C2—22no—
494C2#1397811yescharacteristic
495C2C2×#185611yescharacteristic
496C2C2×#185711yescharacteristic
497C2#2566611yescharacteristic
498C2#2566711yescharacteristic
499C2#2592311yescharacteristic
500C2#2592411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
30. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#342162
C2×#19972
#211712
#207642
#147142
#112962
#108992
#108392
C4×#3392