Finite Group Databasebeta

← #214455

Subgroups — #214455

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic
3#13961C211yesmaximal, characteristic
4#9153C212yesmaximal
5#9145C212yesmaximal
6#9116C214yesmaximal
7#8717C212yesmaximal
8#8333C212yesmaximal
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2855C2212yes—
13#2855C2212yes—
14#2627—21no—
15#2627—21no—
16#2582C414yescomplemented
17#2581—22no—
18#2568C412yescomplemented
19#2567C412yescomplemented
20#2565—22no—
21#2317—21no—
22#2310—21no—
23#2285C2211yescharacteristic
24C2×#383C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1817C2212yes—
29#1817C2212yes—
30C2×#328C2214yes—
31C2×#328C2214yes—
32#1446—21no—
33#1445—21no—
34#1436—21no—
35#1436—21no—
36#1434C2212yes—
37#1434C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×Q8D411yescharacteristic
41#533—42no—
42#533—42no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8—42no—
50C2×C4×Q8—42no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#432—42no—
66#432—42no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×#97—22no—
70#400—22no—
71#400—22no—
72#398—24no—
73#398—24no—
74#395—22no—
75#395—22no—
76#391—22no—
77#391—22no—
78#391—24no—
79#391C2×C414yes—
80#391C2×C414yes—
81#389—24no—
82#389C2×C414yes—
83#389C2×C414yes—
84#387—24no—
85#387—24no—
86#385—22no—
87#385—22no—
88#384—22no—
89#384C2×C412yes—
90#383—42no—
91#382—22no—
92#382—21no—
93#381—22no—
94#381—22no—
95#380—21no—
96#380C2312yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×C412yes—
99#376—42no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95C2312yes—
108C2×#95C2312yes—
109#336—22no—
110#336—22no—
111#336—24no—
112#328—44no—
113#328—44no—
114C22×Q8—21no—
115C22×Q8—24no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—42no—
119C22×Q8C22⋊C412yes—
120C22×Q8C22⋊C412yes—
121C22×Q8C22×C412yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134#125—44no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×Q8—44no—
147#117—44no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C414yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196#97—44no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—22no—
205#95—24no—
206#95—24no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—44no—
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—48no—
223C2×Q8—48no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—82no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4#13714yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C4H(5,𝔽2)12yes—
278C2×C8—48no—
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—82no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C42—24no—
301C42—24no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—44no—
311C42—44no—
312C42—48no—
313C23—22no—
314C23—22no—
315C23C22×C22⋊C411yescharacteristic
316C23#53411yescharacteristic
317C23#53411yescharacteristic
318C23C2×#13712yes—
319C23#57512yes—
320Q8—48no—
321Q8—48no—
322Q8—84no—
323Q8—84no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4#41912yes—
380C2×C4#41912yes—
381C2×C4#51812yes—
382C2×C4#53412yes—
383C8—84no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22#230811yescharacteristic
393C22#288412yes—
394C22C2×#41911yescharacteristic
395C22#289611yescharacteristic
396C22#290112yes—
397C22#319612yes—
398C22#319612yes—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—42no—
408C4—41no—
409C4—84no—
410C4—82no—
411C4—84no—
412C4—82no—
413C4—162no—
414C2—22no—
415C2—22no—
416C2#1762412yes—
417C2#1763212yes—
418C2#2175111yescharacteristic
419C2#2176811yescharacteristic
420C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2582C4semidirect
18. #2568C4semidirect
19. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
#218222
#139612
#91532
#91452
#91162
#87172
#83332