Finite Group Databasebeta

← #364178

Subgroups — #364178

Order 512.

1419 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#1912C212yesmaximal
3#20639C211yesmaximal, characteristic, complemented
4#20355C211yesmaximal, characteristic, complemented
5#11225C212yesmaximal, complemented
6#11056C212yesmaximal
7#10824C212yesmaximal, complemented
8#10775C212yesmaximal
9#10750C212yesmaximal, complemented
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#519C2212yes—
13C2×#519C2211yescharacteristic
14#2905C2212yes—
15C2×#421C2214yes—
16C22×#101C2212yes—
17#2362—22no—
18#2360—22no—
19#2359—24no—
20#2327C412yescomplemented
21#2326C412yescomplemented
22#2317C2211yescharacteristic
23C4×C4⋊Q8C412yescomplemented
24C4×#124C412yescomplemented
25#2297C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1912—28no—
29#1801—22no—
30#1801—22no—
31#1742C2212yes—
32#1740—24no—
33#1740C2212yes—
34#1740C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×H(3,ℤ/4)C2214yes—
38C22×#41C2311yescharacteristic
39C23×Q8C2311yescharacteristic
40C22×M16C2314yes—
41#536—22no—
42#536—22no—
43#536D414yes—
44#536Q814yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#124C2×C412yes—
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49#520—44no—
50#519—28no—
51#519—24no—
52#519—24no—
53#519—44no—
54C2×C4×Q8—44no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×#117—21no—
63C2×#117—24no—
64C2×#117D414yes—
65C2×#117C2314yes—
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#424—44no—
73#423—24no—
74#421—28no—
75#421—28no—
76C2×#104—44no—
77C2×#101—24no—
78C2×#101—24no—
79C2×#101—24no—
80#391—22no—
81#391C2×C411yescharacteristic
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#389—24no—
88#389—22no—
89#388—24no—
90#388—24no—
91#387—24no—
92#387—24no—
93#386—22no—
94#385—24no—
95#385—22no—
96#384C2×C412yes—
97#382C2×C412yes—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—22no—
104#376C2×C412yes—
105C2×#95—22no—
106C43C2×C412yes—
107#343—22no—
108#343—22no—
109#342—24no—
110#342—24no—
111#340—44no—
112#339—28no—
113H(3,ℤ/4)—28no—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—24no—
117C2×#41—24no—
118C2×#41C2×D412yes—
119C2×#41C2×Q812yes—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8C22×C411yescharacteristic
123C22×Q8C22×C412yes—
124C22×Q8C2×D412yes—
125C22×Q8C2×D412yes—
126C22×Q8C2×Q812yes—
127C22×Q8#4112yes—
128C22×Q8#4112yes—
129C22×D4—21no—
130C22×D4C22×C411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16—24no—
143C22×C8—24no—
144C4⋊Q8—24no—
145C4⋊Q8C4⋊C418yes—
146#124—24no—
147#124C4⋊C418yes—
148#123—28no—
149#123—28no—
150C22⋊Q8—28no—
151C22⋊Q8—28no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8—416no—
158C4×Q8—416no—
159C4×D4—44no—
160C4×D4—48no—
161#117—21no—
162#117—21no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—216no—
167#117—24no—
168#117—24no—
169#117—48no—
170#117—44no—
171#117—48no—
172#117C2×D412yes—
173#117C2×Q812yes—
174#117#4118yes—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—44no—
206C2×C42C22⋊C414yes—
207C2×C42C22⋊C414yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C2×D412yes—
213C4⋊C8—48no—
214#104—416no—
215#101—48no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—24no—
222#95—22no—
223C24—21no—
224C24C22×D411yescharacteristic
225#41—28no—
226#41—28no—
227#41—44no—
228#41—44no—
229C2×Q8—21no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—28no—
235C2×Q8—28no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8C2×C4⋊C412yes—
241C2×Q8C2×C4⋊C414yes—
242C2×Q8#11714yes—
243C2×Q8#12114yes—
244C2×Q8C22⋊Q814yes—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4C2×C4⋊C412yes—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4#12114yes—
278C22×C4C22⋊Q814yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4114yes—
286C22×C4H(5,𝔽2)14yes—
287M16—44no—
288M16—44no—
289M16—44no—
290M16—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—216no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—416no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—84no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—216no—
315C22⋊C4—28no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—82no—
320C42—22no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—28no—
326C42—44no—
327C42—44no—
328C42—44no—
329C42—44no—
330C42—416no—
331C42—42no—
332C42—44no—
333C42—82no—
334C42C2×C4⋊C414yes—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23C22×C22⋊C411yescharacteristic
343C23C2×#12111yescharacteristic
344C23C2×C22⋊Q811yescharacteristic
345C23#52512yes—
346C23#53412yes—
347Q8—28no—
348Q8—28no—
349Q8—28no—
350Q8—28no—
351Q8—24no—
352Q8—42no—
353D4—24no—
354D4—44no—
355D4—44no—
356D4—42no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—41no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—164no—
390C2×C4—162no—
391C2×C4#38018yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×#11711yescharacteristic
394C2×C4C4×#4112yes—
395C2×C4#51812yes—
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4#53212yes—
399C2×C4#53412yes—
400C2×C4#54614yes—
401C2×C4#55114yes—
402C2×C4#55314yes—
403C2×C4#55414yes—
404C8—84no—
405C22—21no—
406C22—22no—
407C22—21no—
408C22—22no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—44no—
413C22—42no—
414C22—82no—
415C22—81no—
416C22—81no—
417C22C2×#38011yescharacteristic
418C22#229711yescharacteristic
419C22#230811yescharacteristic
420C22#253412yes—
421C22#257912yes—
422C4—28no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—81no—
431C4—88no—
432C4—82no—
433C4#234912yes—
434C4#235312yes—
435C4#254312yes—
436C4#256412yes—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—22no—
441C2—82no—
442C2#1396011yescharacteristic
443C2#2178311yescharacteristic
444C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20639C2semidirect
4. #20355C2semidirect
5. #11225C2semidirect
7. #10824C2semidirect
9. #10750C2semidirect
20. #2327C4semidirect
21. #2326C4semidirect
23. C4×C4⋊Q8C4semidirect
24. C4×#124C4semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#19122
#206392
#203552
#112252
#110562
#108242
#107752
#107502