Finite Group Databasebeta

← #380298

Subgroups — #380298

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#14697C211yesmaximal, characteristic
6#11328C214yesmaximal, complemented
7#11292C214yesmaximal, complemented
8#10730C211yesmaximal, characteristic
9#10712C211yesmaximal, characteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#389—22no—
18C2×#389—24no—
19C2×#389—24no—
20C2×#389C2214yes—
21C2×#389C2212yes—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C22×#95—21no—
26#1852—24no—
27#1823—24no—
28#1823C418yescomplemented
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#336C2214yes—
35#1436—22no—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C24×C4—21no—
39C24×C4—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—28no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—44no—
55C22×C4⋊C4—44no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4C2×C414yes—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4Q814yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—44no—
71C22×C42D412yes—
72C22×C42C2312yes—
73C2×#107—24no—
74C2×#107—44no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8C2×C414yes—
81#389—216no—
82#389—28no—
83#389—28no—
84#389—44no—
85#385—44no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380C2×C414yes—
90C2×#95—28no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—44no—
101#340C2×C4116yescomplemented
102#336—44no—
103C25C2×D411yescharacteristic
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4D814yes—
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×Q812yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8C22×C414yes—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—216no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—432no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C4⋊C4116yescomplemented
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×Q814yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—48no—
205C2×C42—48no—
206#107—48no—
207C22⋊C8—28no—
208C22⋊C8—44no—
209#95—216no—
210#95—24no—
211#95—44no—
212#95—42no—
213#95—44no—
214#95—44no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24C4×D412yes—
220C24#12012yes—
221C24C22⋊Q812yes—
222C24C2×D812yes—
223C24#13512yes—
224C24AGL(1,ℤ/8)12yes—
225C24#13712yes—
226C24C22×D411yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—28no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—216no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—88no—
299C22×C4—88no—
300C22×C4C2×C4⋊C414yes—
301C22×C4C4×D414yes—
302C22×C4C4×D412yes—
303C22×C4#12012yes—
304C22×C4C22⋊Q814yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12314yes—
307C22×C4C2×D814yes—
308C22×C4#13714yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)14yes—
316C22×C42−1+414yes—
317C2×C8—24no—
318C2×C8—28no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—44no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—216no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—48no—
335C4⋊C4—432no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—88no—
343C4⋊C4C2×C4⋊C418yes—
344C22⋊C4—48no—
345C22⋊C4—84no—
346C42—48no—
347C42—44no—
348C42—44no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—22no—
364C23—22no—
365C23#38514yes—
366C23C4×D814yes—
367C23#44014yes—
368C23#44714yes—
369C23#45214yes—
370C23#47414yes—
371C23#47814yes—
372C23C2×C4×D411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375C23#53412yes—
376C23#53612yes—
377C23C22×D811yescharacteristic
378C23C2×#13511yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380C23C2×#13712yes—
381C23C2×#13711yescharacteristic
382C23#57512yes—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—416no—
399C2×C4—416no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—432no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—816no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#38514yes—
439C2×C4C22×C4⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4#52014yes—
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4#53412yes—
445C2×C4#53512yes—
446C2×C4#53612yes—
447C2×C4#53612yes—
448C2×C4#53914yes—
449C2×C4#53914yes—
450C2×C4#54414yes—
451C8—44no—
452C8—82no—
453C22—28no—
454C22—28no—
455C22—28no—
456C22—28no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22#186218yes—
464C22C2×#38511yescharacteristic
465C22#232212yes—
466C22#261311yescharacteristic
467C22#261411yescharacteristic
468C22#262712yes—
469C22C2×C4×D811yescharacteristic
470C22C2×#44011yescharacteristic
471C22C4×#13712yes—
472C22C2×#44711yescharacteristic
473C22C2×#45211yescharacteristic
474C22#300411yescharacteristic
475C22#300511yescharacteristic
476C22C2×#47411yescharacteristic
477C22C2×#47811yescharacteristic
478C22#307812yes—
479C22#318612yes—
480C22#319012yes—
481C22#319412yes—
482C22#319512yes—
483C22#322612yes—
484C22#323212yes—
485C22#323612yes—
486C22#323812yes—
487C4—24no—
488C4—48no—
489C4—44no—
490C4—44no—
491C4—84no—
492C4—84no—
493C4—161no—
494C4—161no—
495C4#231712yes—
496C4#232212yes—
497C4#262312yes—
498C4#262712yes—
499C2—28no—
500C2#1426311yescharacteristic
501C2C2×#186211yescharacteristic
502C2#2087411yescharacteristic
503C2#2207512yes—
504C2#2208412yes—
505C2#2959211yescharacteristic
506C2#2959911yescharacteristic
507C2#2963811yescharacteristic
508C2#2964311yescharacteristic
509C2#2996612yes—
510C2#2997212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11328C2semidirect
7. #11292C2semidirect
28. #1823C4semidirect
101. #340C2×C4semidirect
185. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#18522
C2×#18232
#146972
#113282
#112922
#107302
#107122