Finite Group Databasebeta

← #481886

Subgroups — #481886

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35748C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#11818C214yesmaximal, complemented
5#11816C214yesmaximal, complemented
6#11723C212yesmaximal, complemented
7#11721C212yesmaximal, complemented
8C2×Q82C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2983—21no—
11#2879—21no—
12#2628C2212yes—
13#2628C2211yescharacteristic
14#2580C411yescharacteristic
15#2580C412yes—
16#2542C411yescharacteristic
17#2542C412yes—
18#2452—21no—
19#2452—22no—
20#2440—22no—
21#2440—21no—
22#2396C2212yes—
23#2396C2211yescharacteristic
24#2351C411yescharacteristic
25Q8×C4⋊C4—21no—
26#2341—21no—
27#1822C2214yes—
28#1821—22no—
29#1742C2212yes—
30C2×#343—21no—
31#1466—22no—
32#1464C2214yes—
33#1460—21no—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2218yes—
37#1353C2214yes—
38C22×#41D411yescharacteristic
39C22×M16—22no—
40Q82—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48#520—44no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D412yes—
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63#446—44no—
64#432—24no—
65#431—22no—
66#408—44no—
67#407—24no—
68#400C2311yescharacteristic
69#400C2312yes—
70#399C2×C412yes—
71#399C2×C412yes—
72#399C2×C412yes—
73#399C2×C412yes—
74#399C2×C411yescharacteristic
75#398—22no—
76#398—21no—
77#396—22no—
78#396—21no—
79#395—22no—
80#395—21no—
81#395C2311yescharacteristic
82#393—22no—
83#393—21no—
84#393C2×C412yes—
85#393C2×C411yescharacteristic
86#390—21no—
87#390—22no—
88#390—21no—
89#390—22no—
90#389—22no—
91#389—21no—
92#389—22no—
93#389—21no—
94#389—21no—
95#389—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384—22no—
100#384—21no—
101#382C2×C412yes—
102#382C2×C411yescharacteristic
103#380—22no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111#343—22no—
112#343—24no—
113#343C2314yes—
114#324C2314yes—
115#323—22no—
116#323—24no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C411yescharacteristic
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C2×M16—24no—
132C2×M16—42no—
133C2×M16—44no—
134C2×M16—44no—
135C22×C8—42no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—48no—
139C4⋊Q8—44no—
140C22⋊Q8—84no—
141C22⋊Q8—84no—
142C22⋊Q8—84no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—44no—
146C4×Q8—42no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×D4—44no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—44no—
153#117—42no—
154#117—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2411yescharacteristic
180C2×C22⋊C4—22no—
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⋊C4C22×C412yes—
188C2×C22⋊C4C22×C411yescharacteristic
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C4⋊C8—42no—
200C4⋊C8—44no—
201C22⋊C8—24no—
202C22⋊C8—48no—
203#97—44no—
204C4×C8—42no—
205#95—21no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—22no—
214#95C22×C411yescharacteristic
215#95C22×C412yes—
216#95C22×C411yescharacteristic
217C24C2×C22⋊C412yes—
218C24C2×C22⋊C411yescharacteristic
219#41—48no—
220#41—48no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8#9914yes—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#13714yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264M16—44no—
265M16—44no—
266M16—84no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—84no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—22no—
305C42—24no—
306C42—24no—
307C42—22no—
308C42—44no—
309C42—44no—
310C42—42no—
311C42—42no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—42no—
319C23—41no—
320C23—42no—
321C23#41912yes—
322C23C22×C22⋊C411yescharacteristic
323Q8—28no—
324Q8—42no—
325Q8—42no—
326Q8—164no—
327Q8—162no—
328D4—48no—
329D4—44no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—48no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—162no—
368C2×C4—162no—
369C2×C4—162no—
370C2×C4C2×#9914yes—
371C2×C4C2×#9912yes—
372C2×C4#41912yes—
373C2×C4#42112yes—
374C2×C4C2×#13712yes—
375C8—84no—
376C8—84no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—42no—
381C22—44no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—81no—
388C22—82no—
389C22C22×#9911yescharacteristic
390C22C2×#41911yescharacteristic
391C22C2×#42111yescharacteristic
392C4—22no—
393C4—24no—
394C4—42no—
395C4—41no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—41no—
402C4—82no—
403C4—81no—
404C4—82no—
405C4—81no—
406C4#151114yes—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—82no—
411C2C2×#151111yescharacteristic
412C2#3334911yescharacteristic
413C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11818C2semidirect
5. #11816C2semidirect
6. #11723C2semidirect
7. #11721C2semidirect

Maximal subgroups

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

subgroupindex
#357482
#206952
#173722
#118182
#118162
#117232
#117212