Finite Group Databasebeta

← #480534

Subgroups — #480534

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11568C214yesmaximal, complemented
5#11565C214yesmaximal, complemented
6#11434C212yesmaximal, complemented
7#11431C212yesmaximal
8C2×#536C411yescharacteristic
9C2×#536C411yescharacteristic
10C22×C4⋊Q8C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2982—21no—
14#2546C412yes—
15#2546C2212yes—
16#2545C412yes—
17#2545C2212yes—
18#2429—21no—
19#2429—21no—
20#2429—21no—
21#2429—21no—
22#2413—22no—
23#2413—22no—
24#2310C2211yescharacteristic
25#2297C2211yescharacteristic
26#2296C411yescharacteristic
27#1467C2214yes—
28#1464—22no—
29#1461—21no—
30#1461—21no—
31#1461C2214yes—
32#1458—22no—
33#1458C2212yes—
34#1458C2212yes—
35#1319C2218yes—
36#1319C2214yes—
37#1318C2214yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41#536—44no—
42#536—44no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#123—22no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58#519—44no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2312yes—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—22no—
71C22×C42C2×C411yescharacteristic
72#446—44no—
73C8×Q8—44no—
74#432—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#396—22no—
83#396—22no—
84#396—22no—
85#396—22no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C411yescharacteristic
90#391C2312yes—
91#391C2311yescharacteristic
92#390C2×C411yescharacteristic
93#390C2311yescharacteristic
94#389—21no—
95#389—21no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#386—21no—
105#386—21no—
106#385—22no—
107#381—21no—
108#381—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#324—24no—
114#324—24no—
115#324C2314yes—
116#324C2314yes—
117#323—22no—
118#323—22no—
119C2×#41—24no—
120C2×#41—42no—
121C2×#41—42no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22⋊C411yescharacteristic
131C22×Q8C22×C412yes—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C22×C412yes—
134C22×D4—21no—
135C22×D4C22×C411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C22⋊C412yes—
140C23×C4C22×C411yescharacteristic
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147#124—44no—
148#124—44no—
149#123—48no—
150#123—48no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—44no—
158C4×Q8—44no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—24no—
162#117—42no—
163#117—44no—
164#117—44no—
165#117—42no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C22⋊C4C2×D411yescharacteristic
198C2×C42—21no—
199C2×C42—42no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C4⋊C8—44no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—44no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216#97—44no—
217C4×C8—44no—
218C4×C8—42no—
219C4×C8—42no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230C24—21no—
231C24C2×C22⋊C411yescharacteristic
232#41—88no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—416no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—88no—
252C2×Q8#10014yes—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#13114yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C2×C8—44no—
290C2×C8—44no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4#9914yes—
310C22⋊C4—24no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23#40812yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—28no—
347Q8—48no—
348Q8—44no—
349Q8—84no—
350Q8—82no—
351Q8—84no—
352D4—82no—
353D4—84no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—28no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—48no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4#40812yes—
393C2×C4C2×#9914yes—
394C2×C4C2×#9912yes—
395C2×C4C2×#10014yes—
396C2×C4C2×#10012yes—
397C2×C4C2×#13112yes—
398C8—84no—
399C8—84no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22C2×#40811yescharacteristic
415C22C22×#9911yescharacteristic
416C22C22×#10011yescharacteristic
417C4—24no—
418C4—22no—
419C4—24no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—41no—
427C4—42no—
428C4—82no—
429C4—81no—
430C4—82no—
431C4#147114yes—
432C2—21no—
433C2—21no—
434C2—42no—
435C2—161no—
436C2C2×#147111yescharacteristic
437C2#3336011yescharacteristic
438C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #17352C2semidirect
4. #11568C2semidirect
5. #11565C2semidirect
6. #11434C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#173712
#173522
#115682
#115652
#114342
#114312