Finite Group Databasebeta

← #379876

Subgroups — #379876

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3#20972C212yesmaximal
4#20688C211yesmaximal, characteristic
5#11225C214yesmaximal, complemented
6#11074C214yesmaximal, complemented
7#10717C212yesmaximal
8C23×M16C2212yes—
9C22×C22⋊Q8C2211yescharacteristic
10C22×#117C2211yescharacteristic
11#2905—24no—
12#2905—22no—
13#2905C2214yes—
14C2×#424—24no—
15C2×#423—22no—
16C2×#419C2214yes—
17#2855—22no—
18#2625—22no—
19#2503C412yescomplemented
20#2502C412yescomplemented
21#2499C414yescomplemented
22#2484—22no—
23#2483—22no—
24#2317C2212yes—
25#2310C2212yes—
26#2280C2212yes—
27#2280C2211yescharacteristic
28#1894—24no—
29#1892—24no—
30#1820—22no—
31#1819—22no—
32#1740C2214yes—
33#1736C2214yes—
34C2×H(3,ℤ/4)C2218yes—
35C23×Q8C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C22×M16—22no—
39C22×M16—22no—
40C22×M16—22no—
41C22×M16—48no—
42C22×M16D414yes—
43C22×M16C2314yes—
44C23×C8D414yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2×C412yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117C2×C412yes—
58C2×#117D412yes—
59C2×#117D412yes—
60C2×#117C2314yes—
61C2×#117C2312yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4D412yes—
69C22×C42D412yes—
70#424—216no—
71#424—216no—
72#423—28no—
73#423—28no—
74C2×C4⋊C8—28no—
75#419—48no—
76C2×#103—28no—
77#400—24no—
78#398—24no—
79#397—28no—
80#391—24no—
81#391C2×C414yes—
82#391C2×C414yes—
83#389—24no—
84#387—28no—
85#385—24no—
86#385—24no—
87#384C2×C414yes—
88#382C2×C414yes—
89#381—24no—
90#381—21no—
91#381—24no—
92#381—22no—
93#380—21no—
94#380—21no—
95#380—22no—
96#380—22no—
97#379—21no—
98#379—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4C2×C412yes—
103#342—28no—
104#340—28no—
105H(3,ℤ/4)—48no—
106C25C2×D411yescharacteristic
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8C22×C412yes—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4C2411yescharacteristic
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—28no—
134C2×M16—28no—
135C2×M16—48no—
136C2×M16—48no—
137C2×M16—48no—
138C2×M16—48no—
139C2×M16—48no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—48no—
143C22×C8C2×D418yes—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—42no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—42no—
154#117—24no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—28no—
165#117—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4C22⋊C414yes—
198C2×C22⋊C4C2×D414yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42C22⋊C414yes—
205C2×C42C22×C414yes—
206C2×C42C2×D414yes—
207C4⋊C8—232no—
208#103—416no—
209#95—22no—
210#95—22no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—42no—
215#95—42no—
216C24—24no—
217C24—42no—
218C24—81no—
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C411yescharacteristic
221C24#12112yes—
222C24#12112yes—
223C24C22×D411yescharacteristic
224C2×Q8—24no—
225C2×Q8—28no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12012yes—
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12311yescharacteristic
274C22×C4#12311yescharacteristic
275C22×C4#12312yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)14yes—
281C22×C42−1+414yes—
282M16—48no—
283M16—48no—
284M16—48no—
285M16—48no—
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—28no—
289C2×C8—48no—
290C2×C8—48no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—88no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—48no—
314C22⋊C4—82no—
315C22⋊C4—84no—
316C22⋊C4—84no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—28no—
322C42—48no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—24no—
327C23—22no—
328C23—24no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23—41no—
336C23—42no—
337C23—88no—
338C23#38612yes—
339C23#38611yescharacteristic
340C23#38611yescharacteristic
341C23C22×C22⋊C411yescharacteristic
342C23C2×#12112yes—
343C23#53714yes—
344Q8—48no—
345Q8—81no—
346Q8—81no—
347Q8—82no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38612yes—
388C2×C4C2×C4×D412yes—
389C2×C4#52014yes—
390C2×C4C2×#12011yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4#53414yes—
393C2×C4#53512yes—
394C2×C4#53614yes—
395C2×C4#53612yes—
396C2×C4#53918yes—
397C8—48no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—21no—
410C22—44no—
411C22—44no—
412C22—48no—
413C22—82no—
414C22—84no—
415C22#190912yes—
416C22C2×#38611yescharacteristic
417C22#231112yes—
418C22#261814yes—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—44no—
423C4—44no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—164no—
428C4#232214yes—
429C4#261311yescharacteristic
430C4#261411yescharacteristic
431C4#262312yes—
432C2—22no—
433C2—24no—
434C2—22no—
435C2—41no—
436C2—42no—
437C2#1425611yescharacteristic
438C2C2×#190911yescharacteristic
439C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. #11225C2semidirect
6. #11074C2semidirect
19. #2503C4semidirect
20. #2502C4semidirect
21. #2499C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#353112
#209722
#206882
#112252
#110742
#107172