Finite Group Databasebeta

← #482140

Subgroups — #482140

Order 512.

949 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35748C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#11901C214yesmaximal, complemented
5#11897C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11715C212yesmaximal, complemented
8C2×Q82C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2972—21no—
11C2×#410—21no—
12#2628C412yescomplemented
13#2628C2211yescharacteristic
14#2580C411yescharacteristic, complemented
15#2580C2212yes—
16#2542C411yescharacteristic, complemented
17#2542C2212yes—
18#2452—22no—
19#2452—21no—
20#2440—21no—
21#2440—22no—
22#2396C412yescomplemented
23#2396C2211yescharacteristic
24#2351C411yescharacteristic, complemented
25Q8×C4⋊C4—21no—
26#2341—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1352C2218yes—
36#1352C2214yes—
37#1351C2214yes—
38C22×#41D411yescharacteristic
39Q82—44no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47#520—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D412yes—
63C8×D4—44no—
64#431—22no—
65#410—44no—
66C2×#99—24no—
67C2×C22⋊C8—24no—
68#400C2×C411yescharacteristic
69#400C2×C412yes—
70#399C2×C412yes—
71#399C2×C412yes—
72#399C2×C412yes—
73#399C2312yes—
74#399C2311yescharacteristic
75#398—22no—
76#398—21no—
77#396—22no—
78#396—21no—
79#395—22no—
80#395—21no—
81#395C2×C411yescharacteristic
82#393—22no—
83#393—21no—
84#393C2×C412yes—
85#393C2311yescharacteristic
86#390—21no—
87#390—21no—
88#390—22no—
89#390—22no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389—21no—
96#389—44no—
97#385—22no—
98#385—22no—
99#385—22no—
100#384—21no—
101#384—22no—
102#382C2×C411yescharacteristic
103#382C2×C412yes—
104#380—22no—
105#380—21no—
106#380—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41SD1614yes—
121C2×#41Q1614yes—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×D4C22⋊C411yescharacteristic
127C22×D4C2×D412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C22×C8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—42no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
138C22⋊Q8—84no—
139C22⋊Q8—84no—
140C22⋊Q8—84no—
141C4×Q8—24no—
142C4×Q8—42no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×D4—24no—
148C4×D4—42no—
149C4×D4—44no—
150C4×D4—44no—
151#117—42no—
152#117—44no—
153#117—44no—
154#117—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C412yes—
192C2×C22⋊C4C2×D411yescharacteristic
193C2×C22⋊C4C2411yescharacteristic
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—42no—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C4⋊C8—42no—
204#99—48no—
205C22⋊C8—24no—
206C22⋊C8—44no—
207C4×C8—42no—
208#95—21no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—44no—
218#95C22×C411yescharacteristic
219#95C22×C412yes—
220#95C22×C411yescharacteristic
221C24C2×C22⋊C412yes—
222C24C22×D411yescharacteristic
223#41—48no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—44no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8#10314yes—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4#10314yes—
240C2×D4C2×SD1614yes—
241C2×D4C2×Q1614yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4#10318yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×SD1612yes—
268C22×C4C2×Q1612yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—44no—
272C2×C8—164no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—84no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—164no—
297C4⋊C4#9914yes—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C42—22no—
312C42—24no—
313C42—22no—
314C42—42no—
315C42—44no—
316C42—44no—
317C42—42no—
318C42—42no—
319C42—84no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—42no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23C2×#10314yes—
329C23C22×C22⋊C411yescharacteristic
330C23C22×SD1611yescharacteristic
331C23C22×Q1611yescharacteristic
332Q8—28no—
333Q8—42no—
334Q8—42no—
335Q8—164no—
336Q8—162no—
337D4—28no—
338D4—44no—
339D4—44no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—48no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—162no—
382C2×C4C2×#9914yes—
383C2×C4C2×#9912yes—
384C2×C4C2×#10312yes—
385C2×C4C2×#10314yes—
386C2×C4C2×#10312yes—
387C2×C4#42112yes—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—24no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—42no—
401C22—81no—
402C22—82no—
403C22C22×#9911yescharacteristic
404C22C22×#10311yescharacteristic
405C22C2×#42111yescharacteristic
406C4—24no—
407C4—22no—
408C4—24no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4—82no—
420C4—164no—
421C4#152014yes—
422C2—21no—
423C2—21no—
424C2—41no—
425C2—41no—
426C2—44no—
427C2C2×#152011yescharacteristic
428C2#3334911yescharacteristic
429C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11901C2semidirect
5. #11897C2semidirect
6. #11718C2semidirect
7. #11715C2semidirect
8. C2×Q82C4semidirect
12. #2628C4semidirect
14. #2580C4semidirect
16. #2542C4semidirect
22. #2396C4semidirect
24. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#357482
#206342
#173512
#119012
#118972
#117182
#117152