Finite Group Databasebeta

← #214888

Subgroups — #214888

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#9488C212yesmaximal, complemented
5#9400C212yesmaximal, complemented
6#9192C212yesmaximal, complemented
7#9034C212yesmaximal, complemented
8#8765C212yesmaximal, complemented
9#8391C212yesmaximal, complemented
10C2×D4×Q8C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2921C2212yes—
14#2628C2211yescharacteristic
15#2627C412yes—
16#2581C2211yescharacteristic
17#2567C2211yescharacteristic
18#2564C412yes—
19#2451—21no—
20#2449—22no—
21#2440—21no—
22#2438—22no—
23#2409C412yes—
24#2396C2211yescharacteristic
25C2×#395C412yes—
26#2349C2211yescharacteristic
27D4×C4⋊C4—21no—
28#2339—21no—
29#1821—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1466—22no—
38#1464—22no—
39#1460C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C23×Q8C2×C412yes—
45D4×Q8—44no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51#520—44no—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D412yes—
60C2×C4×D4C2311yescharacteristic
61C2×#117D412yes—
62C2×#117C2311yescharacteristic
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66#446—44no—
67#433—44no—
68#432—24no—
69#431C2314yes—
70#423C2314yes—
71C2×#101—22no—
72C2×#100—22no—
73#407—22no—
74C4×M16—22no—
75#400—21no—
76#400C2×C412yes—
77#400C2×C411yescharacteristic
78#399—21no—
79#399C2×C411yescharacteristic
80#398—22no—
81#398—22no—
82#398—21no—
83#396—21no—
84#395—22no—
85#395—22no—
86#395C2×C412yes—
87#395C2×C412yes—
88#395C2×C412yes—
89#395C2×C412yes—
90#395C2×C411yescharacteristic
91#393C2×C412yes—
92#393C2×C411yescharacteristic
93#391—21no—
94#391—21no—
95#391—22no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389—21no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#382C2311yescharacteristic
110#381—22no—
111#380—21no—
112#380—22no—
113#380C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2311yescharacteristic
116C4×C22⋊C4—21no—
117C2×#95—22no—
118C2×#95C2×C412yes—
119#343—22no—
120#343—22no—
121#333—44no—
122Q8⋊C8—44no—
123#324—22no—
124#323—22no—
125#323—24no—
126C2×#41—24no—
127C2×#41—42no—
128C2×#41—42no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8—84no—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C22×C412yes—
134C22×Q8C2×D411yescharacteristic
135C22×D4C22⋊C412yes—
136C22×D4C2×D411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C4⋊Q8—24no—
146C4⋊Q8—44no—
147C22⋊Q8—48no—
148C22⋊Q8—44no—
149C22⋊Q8—84no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—42no—
153C4×D4—24no—
154C4×D4—44no—
155C4×D4—42no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—44no—
159#117—24no—
160#117—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—84no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22×C411yescharacteristic
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22⋊C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—44no—
204#101—44no—
205#100—44no—
206C22⋊C8—24no—
207C22⋊C8—44no—
208#97—42no—
209#97—44no—
210C4×C8—24no—
211C4×C8—42no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95C22×C411yescharacteristic
226#95C22×C412yes—
227#95C22×C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C24C2×C22⋊C412yes—
230#41—416no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4#13714yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—44no—
283M16—44no—
284C2×C8—44no—
285C2×C8—44no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C42—22no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42H(5,𝔽2)14yes—
322C23—22no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23#41914yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—24no—
334Q8—42no—
335Q8—44no—
336Q8—84no—
337Q8—88no—
338D4—28no—
339D4—48no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—48no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4—162no—
380C2×C4—162no—
381C2×C4#51812yes—
382C2×C4#53412yes—
383C2×C4#53412yes—
384C2×C4C2×#13712yes—
385C2×C4#57512yes—
386C8—84no—
387C8—84no—
388C22—21no—
389C22—21no—
390C22—24no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22#230811yescharacteristic
401C22C2×#41911yescharacteristic
402C22#289611yescharacteristic
403C4—22no—
404C4—22no—
405C4—42no—
406C4—41no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—41no—
413C4—81no—
414C4—82no—
415C4—82no—
416C4—81no—
417C4#319614yes—
418C2—21no—
419C2—21no—
420C2—44no—
421C2—81no—
422C2#2176811yescharacteristic
423C2#2177811yescharacteristic
424C2#2178311yescharacteristic
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. #9488C2semidirect
5. #9400C2semidirect
6. #9192C2semidirect
7. #9034C2semidirect
8. #8765C2semidirect
9. #8391C2semidirect

Maximal subgroups

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

subgroupindex
#357392
#207312
#173722
#94882
#94002
#91922
#90342
#87652
#83912