Finite Group Databasebeta

← #379873

Subgroups — #379873

Order 512.

1629 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#35227C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#20979C212yesmaximal
4C2×#1821C211yesmaximal, characteristic, complemented
5#11224C214yesmaximal, complemented
6#11072C214yesmaximal, complemented
7#10716C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#421C2214yes—
12C2×#407C2212yes—
13#2849—22no—
14#2623—22no—
15#2504C412yescomplemented
16#2503C412yescomplemented
17#2502C412yescomplemented
18#2501C412yescomplemented
19#2487—22no—
20#2486—22no—
21#2317C2212yes—
22#2316—24no—
23#2316—22no—
24#2316C2214yes—
25#2311C2212yes—
26C2×#389—24no—
27#2280C2211yescharacteristic
28C2×#380—22no—
29#1897—24no—
30#1896—24no—
31#1821—22no—
32#1821—22no—
33C2×#343C2214yes—
34#1733C2214yes—
35C2×H(3,ℤ/4)C2218yes—
36C22×#41C2311yescharacteristic
37C24×C4D412yes—
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40#525—22no—
41#525—22no—
42#525—22no—
43#525—22no—
44C2×#123—21no—
45C2×#123—24no—
46C2×C22⋊Q8C2×C411yescharacteristic
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×#121C2×C411yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#120—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—48no—
56C2×#117D414yes—
57C2×#117C2311yescharacteristic
58C2×#117C2312yes—
59C2×#117C2314yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C42D414yes—
74#421—48no—
75C2×#104—28no—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82#400—24no—
83#399—24no—
84#398—24no—
85#397—24no—
86#391C2×C414yescomplemented
87#390C2×C414yescomplemented
88#389—216no—
89#389—24no—
90#389—216no—
91#386—24no—
92#386—24no—
93#385—24no—
94#385—24no—
95#384C2×C414yescomplemented
96#382C2×C414yescomplemented
97#381—24no—
98#381—21no—
99#380—28no—
100#380—28no—
101#380—21no—
102#380—21no—
103#380—24no—
104#379—21no—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C412yes—
107C2×#95—24no—
108#343—28no—
109#339—28no—
110H(3,ℤ/4)—48no—
111C25C2×D411yescharacteristic
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—24no—
115C2×#41—24no—
116C22×Q8C22×C411yescharacteristic
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C22×C411yescharacteristic
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—84no—
130C23×C4C22⋊C414yes—
131C23×C4C22⋊C414yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D414yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—28no—
146C2×M16—28no—
147C22×C8—24no—
148#123—42no—
149#123—42no—
150#123—88no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121—22no—
159#120—41no—
160#120—41no—
161#120—41no—
162#120—41no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167#117—28no—
168#117—28no—
169#117—24no—
170#117—24no—
171#117—22no—
172#117—22no—
173#117—48no—
174#117—48no—
175#117—48no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—88no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22⋊C414yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—48no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—48no—
218C2×C42C22×C414yes—
219C2×C42C2×D418yes—
220#104—416no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#95—216no—
226#95—22no—
227#95—22no—
228#95—28no—
229#95—28no—
230C24—22no—
231C24—22no—
232C24—21no—
233C24—42no—
234C24—42no—
235C24C2×C22⋊C411yescharacteristic
236C24C2×C22⋊C411yescharacteristic
237C24#12011yescharacteristic
238C24#12011yescharacteristic
239C24#12311yescharacteristic
240C24#12311yescharacteristic
241C24C22×D411yescharacteristic
242#41—44no—
243#41—44no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—28no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C418yes—
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12012yes—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4H(5,𝔽2)14yes—
312C22×C42−1+414yes—
313M16—44no—
314M16—44no—
315M16—48no—
316C2×C8—44no—
317C2×C8—48no—
318C4⋊C4—28no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—816no—
330C4⋊C4—88no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—48no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—88no—
346C42—28no—
347C42—28no—
348C42—28no—
349C42—48no—
350C42—42no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—48no—
374C23—48no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23#38611yescharacteristic
380C23#38611yescharacteristic
381C23#38611yescharacteristic
382C23#38611yescharacteristic
383C23C22×C22⋊C411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×#12311yescharacteristic
386C23#53512yes—
387C23#53612yes—
388Q8—44no—
389D4—44no—
390D4—82no—
391D4—82no—
392D4—81no—
393D4—81no—
394D4—82no—
395D4—82no—
396D4—82no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—28no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—816no—
431C2×C4—84no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4C2×C4×D412yes—
435C2×C4#52014yes—
436C2×C4C2×#12112yes—
437C2×C4#53414yes—
438C2×C4#53614yes—
439C2×C4#53714yes—
440C2×C4#53918yes—
441C8—84no—
442C22—24no—
443C22—24no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—84no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#191112yes—
473C22#191212yes—
474C22C2×#38611yescharacteristic
475C22#231112yes—
476C22#261311yescharacteristic
477C22#261411yescharacteristic
478C22#262312yes—
479C4—22no—
480C4—22no—
481C4—48no—
482C4—84no—
483C4—88no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4#232214yes—
488C4#261814yes—
489C2—21no—
490C2—22no—
491C2—21no—
492C2—22no—
493C2—22no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—161no—
498C2#1425611yescharacteristic
499C2C2×#191111yescharacteristic
500C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2316C2semidirect
4. C2×#1821C2semidirect
5. #11224C2semidirect
6. #11072C2semidirect
7. #10716C2semidirect
15. #2504C4semidirect
16. #2503C4semidirect
17. #2502C4semidirect
18. #2501C4semidirect
86. #391C2×C4semidirect
87. #390C2×C4semidirect
95. #384C2×C4semidirect
96. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352272
C2×#23162
#209792
C2×#18212
#112242
#110722
#107162