Finite Group Databasebeta

← #214845

Subgroups — #214845

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36040C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#9483C212yesmaximal, complemented
5#9391C212yesmaximal, complemented
6#9192C212yesmaximal, complemented
7#9034C212yesmaximal, complemented
8#8766C212yesmaximal, complemented
9#8392C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2972C2212yes—
13#2921C2212yes—
14#2628C412yes—
15#2628C2211yescharacteristic
16#2580C412yes—
17#2580C2211yescharacteristic
18#2563C412yes—
19#2563C2211yescharacteristic
20#2454—22no—
21#2454—21no—
22#2442—21no—
23#2442—22no—
24#2396C412yes—
25#2396C2211yescharacteristic
26#2351—21no—
27#2351C2211yescharacteristic
28#2341—21no—
29#1821—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1461C2212yes—
38#1459—22no—
39#1458C2212yes—
40C2×#324—22no—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44#557—44no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—22no—
48C2×#125—21no—
49#520—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117C2×C412yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C8×D4—44no—
65#433—44no—
66#431C2314yes—
67#423C2314yes—
68C2×#101—22no—
69C2×#100—22no—
70#407—22no—
71C2×C22⋊C8—24no—
72C4×M16—22no—
73#400—22no—
74#400—21no—
75#400C2×C411yescharacteristic
76#399—22no—
77#399—21no—
78#399C2×C411yescharacteristic
79#399C2×C412yes—
80#398—22no—
81#398—21no—
82#396—22no—
83#396—21no—
84#395C2×C411yescharacteristic
85#395C2×C412yes—
86#393C2×C411yescharacteristic
87#393C2×C412yes—
88#393C2×C412yes—
89#393C2×C412yes—
90#393C2×C412yes—
91#390—22no—
92#390—22no—
93#390—21no—
94#390—21no—
95#389—22no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389—22no—
100#389—21no—
101#385—22no—
102#385—22no—
103#385—22no—
104#382—22no—
105#382—21no—
106#382C2×C412yes—
107#382C2311yescharacteristic
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119#343—22no—
120#343—22no—
121#333—44no—
122Q8⋊C8—44no—
123#324—22no—
124#324—24no—
125#323—22no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C22×Q8C22×C412yes—
130C22×Q8C22×C411yescharacteristic
131C22×Q8C2×D411yescharacteristic
132C22×D4C22⋊C412yes—
133C22×D4C2×D411yescharacteristic
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C2×M16—24no—
140C22×C8—44no—
141C4⋊Q8—24no—
142C4⋊Q8—44no—
143#125—48no—
144#125—44no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—44no—
148C4×Q8—42no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×D4—24no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—44no—
155#117—24no—
156#117—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C22×C411yescharacteristic
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22⋊C412yes—
197C2×C42C22×C412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—24no—
203#101—44no—
204#100—44no—
205C22⋊C8—24no—
206C22⋊C8—44no—
207C22⋊C8—44no—
208#97—42no—
209C4×C8—24no—
210C4×C8—42no—
211#95—21no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C412yes—
227#41—416no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—81no—
233C2×Q8—82no—
234C2×Q8#13514yes—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4#13714yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268M16—44no—
269M16—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—164no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—81no—
294C4⋊C4—82no—
295C22⋊C4—22no—
296C22⋊C4—24no—
297C22⋊C4—22no—
298C22⋊C4—42no—
299C22⋊C4—82no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—24no—
307C42—22no—
308C42—42no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—44no—
313C42H(5,𝔽2)14yes—
314C23—22no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23—41no—
323C23#41914yes—
324C23C22×C22⋊C411yescharacteristic
325Q8—28no—
326Q8—44no—
327Q8—84no—
328D4—28no—
329D4—48no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—84no—
366C2×C4—162no—
367C2×C4—162no—
368C2×C4#41614yes—
369C2×C4#51812yes—
370C2×C4#53412yes—
371C2×C4#53412yes—
372C2×C4C2×#13512yes—
373C2×C4C2×#13712yes—
374C8—84no—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—82no—
387C22—81no—
388C22#230811yescharacteristic
389C22C2×#41611yescharacteristic
390C22C2×#41911yescharacteristic
391C4—24no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—41no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4—81no—
403C4—82no—
404C4—81no—
405C4#319014yes—
406C2—21no—
407C2—21no—
408C2—44no—
409C2—81no—
410C2#2176211yescharacteristic
411C2#2178111yescharacteristic
412C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9483C2semidirect
5. #9391C2semidirect
6. #9192C2semidirect
7. #9034C2semidirect
8. #8766C2semidirect
9. #8392C2semidirect

Maximal subgroups

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

subgroupindex
#360402
#207312
#173512
#94832
#93912
#91922
#90342
#87662
#83922