Finite Group Databasebeta

← #382821

Subgroups — #382821

Order 512.

1203 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20634C211yesmaximal, characteristic
4#11224C212yesmaximal, complemented
5#11160C212yesmaximal
6#11055C212yesmaximal, complemented
7#10988C212yesmaximal
8#10816C212yesmaximal, complemented
9#10777C212yesmaximal, complemented
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#519C2211yescharacteristic
13C2×#419C2214yes—
14C2×#410C2212yes—
15#2879C2212yes—
16#2362—22no—
17#2360—22no—
18#2359—24no—
19#2327C412yescomplemented
20#2326C412yescomplemented
21#2317C2211yescharacteristic
22#2316C2212yes—
23C4×C4⋊Q8C412yescomplemented
24C4×#124C412yescomplemented
25#2297C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1822—22no—
29#1821—22no—
30#1799—24no—
31#1742C2212yes—
32C2×#343C2212yes—
33#1740C2212yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40#536—24no—
41#536—24no—
42C2×C4⋊Q8C2×C412yes—
43C2×#124C2×C412yes—
44C2×#123—22no—
45C2×C22⋊Q8—22no—
46#520—44no—
47#519—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×D4C2×C412yes—
54C2×C4×D4D414yes—
55C2×#117—22no—
56C2×#117D412yes—
57C2×#117C2314yes—
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#419—28no—
65#419—44no—
66C2×#103—44no—
67#410—44no—
68C2×#99—24no—
69#408—44no—
70#407—24no—
71#391—22no—
72#391C2×C411yescharacteristic
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C411yescharacteristic
76#390C2×C411yescharacteristic
77#390C2×C411yescharacteristic
78#389—24no—
79#389—22no—
80#389—44no—
81#388—24no—
82#388—24no—
83#387—24no—
84#387—24no—
85#386—22no—
86#385—24no—
87#385—22no—
88#384C2×C412yes—
89#382C2×C412yes—
90#380—24no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#376C2×C412yes—
98C2×#95—22no—
99C43C2×C412yes—
100#343—22no—
101#343—24no—
102#343—22no—
103#342—24no—
104#339—44no—
105#339—44no—
106H(3,ℤ/4)—28no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—24no—
110C2×#41—22no—
111C2×#41—22no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8C22×C412yes—
118C22×Q8C22×C411yescharacteristic
119C22×Q8#4114yes—
120C22×D4C22×C411yescharacteristic
121C22×D4C2×D412yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C22⋊C414yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C22×C8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137#124—24no—
138#124—24no—
139#123—216no—
140C22⋊Q8—216no—
141C4×Q8—44no—
142C4×Q8—44no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—44no—
146C4×D4—44no—
147C4×D4—48no—
148#117—22no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153#117—24no—
154#117—24no—
155#117—48no—
156#117—44no—
157#117#4114yes—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C22⋊C4—24no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42C22⋊C414yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196#103—416no—
197#99—48no—
198C22⋊C8—48no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—24no—
205#95—22no—
206#95—44no—
207C24C2×C22⋊C412yes—
208C24C22×D411yescharacteristic
209#41—44no—
210#41—44no—
211#41—44no—
212#41—44no—
213C2×Q8—22no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—28no—
222C2×Q8—28no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8#11714yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4112yes—
265C22×C4C2×#4112yes—
266C22×C4H(5,𝔽2)14yes—
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269M16—44no—
270M16—44no—
271M16—44no—
272M16—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—216no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—84no—
290C4⋊C4—164no—
291C22⋊C4—216no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—82no—
300C42—22no—
301C42—24no—
302C42—22no—
303C42—22no—
304C42—44no—
305C42—44no—
306C42—44no—
307C42—44no—
308C42—42no—
309C42—42no—
310C42—44no—
311C42—44no—
312C42—82no—
313C42#11714yes—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—42no—
321C23C22×C22⋊C411yescharacteristic
322C23#52512yes—
323C23#53412yes—
324C23#53511yescharacteristic
325C23#53611yescharacteristic
326Q8—28no—
327Q8—42no—
328Q8—44no—
329Q8—42no—
330Q8—48no—
331D4—44no—
332D4—44no—
333D4—42no—
334D4—42no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—164no—
370C2×C4C2×#11711yescharacteristic
371C2×C4C2×#11711yescharacteristic
372C2×C4#51812yes—
373C2×C4#52012yes—
374C2×C4#52512yes—
375C2×C4#53412yes—
376C2×C4#53712yes—
377C2×C4#53814yes—
378C2×C4#53914yes—
379C2×C4#54014yes—
380C2×C4#54214yes—
381C8—84no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22#228011yescharacteristic
395C22#230811yescharacteristic
396C22#231111yescharacteristic
397C22#263512yes—
398C22#263812yes—
399C4—44no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—42no—
404C4—44no—
405C4—42no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—164no—
410C4#231812yes—
411C4#231912yes—
412C4#263014yes—
413C2—21no—
414C2—21no—
415C2—22no—
416C2—22no—
417C2—82no—
418C2#1434411yescharacteristic
419C2#2178311yescharacteristic
420C2#2178411yescharacteristic
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
4. #11224C2semidirect
6. #11055C2semidirect
8. #10816C2semidirect
9. #10777C2semidirect
19. #2327C4semidirect
20. #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
#206952
#206342
#112242
#111602
#110552
#109882
#108162
#107772