Finite Group Databasebeta

← #495175

Subgroups — #495175

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2C2×#2129C214yesmaximal
3#20632C211yesmaximal, characteristic
4C4×#410C211yesmaximal, characteristic
5#13224C218yesmaximal, complemented
6C2×#536C2212yes—
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#410C2212yes—
10C2×#410C2212yes—
11#2857—21no—
12#2857—21no—
13#2621—21no—
14#2621—21no—
15#2547C2214yes—
16#2546C414yescomplemented
17#2486—22no—
18#2486—22no—
19#2310C412yescomplemented
20#2285—21no—
21#2285C411yescharacteristic, complemented
22C4×#117—21no—
23#2129—216no—
24C2×#354C22116yes—
25#1799—22no—
26#1799C2214yes—
27C4×#99—22no—
28C4×#99C2214yes—
29#1740—21no—
30#1740—21no—
31C22×#41C2311yescharacteristic
32C23×Q8C2×C412yes—
33#536—28no—
34#536—28no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—42no—
39C2×C4⋊Q8—41no—
40C2×C4⋊Q8C2×C414yes—
41C2×C4⋊Q8C2×C414yes—
42C2×#124C2314yes—
43C2×#123—22no—
44C2×#123—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C4×#41—41no—
48C4×#41—41no—
49C4×#41—42no—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C412yes—
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C22×C4⋊C4—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64#410—24no—
65#410—28no—
66#410D418yes—
67C2×#99—22no—
68C2×#99—22no—
69C2×#99C2314yes—
70C2×#99C2314yes—
71#398—24no—
72#398—24no—
73#397—24no—
74#397—24no—
75#391—22no—
76#391—22no—
77#391C2×C418yes—
78#389—24no—
79#389—24no—
80#384—21no—
81#384—41no—
82#382—21no—
83#382—41no—
84#381—24no—
85#380—22no—
86#380—24no—
87#380—24no—
88#380—24no—
89C4×C4⋊C4—42no—
90C4×C4⋊C4—42no—
91C4×C4⋊C4C2×C414yes—
92C4×C22⋊C4—24no—
93#376—41no—
94C43—41no—
95#354—232no—
96#342—24no—
97#342—24no—
98C2×#41—41no—
99C2×#41—41no—
100C2×#41—42no—
101C2×#41C2×D414yes—
102C2×#41#4114yes—
103C22×Q8—22no—
104C22×Q8—28no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—44no—
108C22×Q8C22×C414yes—
109C22×Q8C22×C411yescharacteristic
110C22×D4C2×D411yescharacteristic
111C22×D4C2×D411yescharacteristic
112C22×D4C2411yescharacteristic
113C23×C4—22no—
114C23×C4C22⋊C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C4⋊Q8—216no—
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—41no—
126C4⋊Q8—42no—
127C4⋊Q8—41no—
128#124—28no—
129#124—28no—
130#123—48no—
131#123—44no—
132#123—44no—
133C22⋊Q8—48no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144#117—28no—
145#117—28no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—44no—
153#117—44no—
154#117—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C22⋊C4C2×D412yes—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—44no—
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22×C414yes—
196C2×C42C22×C412yes—
197#99—28no—
198#99—24no—
199#99—28no—
200#99—44no—
201#99C2×D418yes—
202#95—24no—
203#95—28no—
204#95—28no—
205#95—44no—
206#95—44no—
207C24C2×C22⋊C411yescharacteristic
208C24C22×D411yescharacteristic
209C24C22×D411yescharacteristic
210#41—44no—
211#41—44no—
212#41—44no—
213#41—44no—
214C2×Q8—28no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—88no—
224C2×Q8—88no—
225C2×Q8C4×D414yes—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#12114yes—
233C2×D4#12412yes—
234C2×D4#12712yes—
235C2×D4C22×D412yes—
236C2×D4C2×#4112yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
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—24no—
253C22×C4—24no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C4×D414yes—
270C22×C4#12114yes—
271C22×C4#12412yes—
272C22×C4#12712yes—
273C22×C4C23×C411yescharacteristic
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—84no—
286C4⋊C4—88no—
287C22⋊C4—28no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—48no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—28no—
311C42—48no—
312C42—84no—
313C42—88no—
314C42#9911yescharacteristic
315C42#9912yes—
316C42#9911yescharacteristic
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23#39012yes—
327C23C22×C22⋊C411yescharacteristic
328C23C2×#12112yes—
329C23C2×#12411yescharacteristic
330C23C2×#12711yescharacteristic
331Q8—44no—
332Q8—88no—
333Q8—84no—
334Q8—84no—
335D4—44no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—84no—
376C2×C4—88no—
377C2×C4#39012yes—
378C2×C4#39012yes—
379C2×C4#39012yes—
380C2×C4C2×#9912yes—
381C2×C4C2×#9912yes—
382C2×C4C2×#9911yescharacteristic
383C2×C4C2×#9911yescharacteristic
384C2×C4#41012yes—
385C2×C4C2×C4×D412yes—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—81no—
400C22—81no—
401C22C2×#39011yescharacteristic
402C22C22×#9911yescharacteristic
403C22C2×#41011yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—84no—
412C4—88no—
413C4—84no—
414C4—164no—
415C4#196412yes—
416C4#196412yes—
417C4#212914yescomplemented
418C2—21no—
419C2—21no—
420C2—42no—
421C2—41no—
422C2—42no—
423C2—41no—
424C2C2×#196411yescharacteristic
425C2C2×#212912yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13224C2semidirect
7. C22×C4⋊Q8C4semidirect
16. #2546C4semidirect
19. #2310C4semidirect
21. #2285C4semidirect
417. C4#2129semidirect

Maximal subgroups

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

subgroupindex
#353162
C2×#21292
#206322
C4×#4102
#132242