Finite Group Databasebeta

← #190842

Subgroups — #190842

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#21431C211yesmaximal, characteristic
4#8708C212yesmaximal
5#8570C212yesmaximal, complemented
6#8568C212yesmaximal, complemented
7#8311C212yesmaximal
8#8182C212yesmaximal, complemented
9#8180C212yesmaximal, complemented
10C22×C4×Q8C2211yescharacteristic
11#2320—22no—
12#2320C412yes—
13#2320C412yes—
14#2316C411yescharacteristic
15#2316C411yescharacteristic
16#2313—22no—
17#2310C411yescharacteristic
18#2310C411yescharacteristic
19#2295—22no—
20#2295C2212yes—
21#2295C2212yes—
22#2283—22no—
23C4×#117C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1917—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1831C2211yescharacteristic
31#1797C2212yes—
32C4×C4⋊C8C2212yes—
33C4×C4⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C4×#97C2211yescharacteristic
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C23×Q8C2×C411yescharacteristic
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2312yes—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70#391—22no—
71#391—22no—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#389—22no—
77#389—22no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#383—22no—
91#383—22no—
92#382—22no—
93#382—22no—
94#381—22no—
95#381—22no—
96#380—22no—
97#380—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4D412yes—
107C4×C22⋊C4D412yes—
108#376—21no—
109#376—22no—
110#376—22no—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43—21no—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8C22×C412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4×Q8—28no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8#4114yes—
148C4×Q8#4114yes—
149#117—24no—
150#117—24no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—44no—
198C4⋊C8—82no—
199C22⋊C8—44no—
200C22⋊C8—82no—
201#97—44no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—82no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225C24C2×C22⋊C411yescharacteristic
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—416no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8#11714yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—82no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4#13114yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—24no—
275C4⋊C4—28no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4#11714yes—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—28no—
293C42—22no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—82no—
314C42C2×#4112yes—
315C42C2×#4112yes—
316C23—21no—
317C23—22no—
318C23—22no—
319C23—21no—
320C23#40812yes—
321C23C22×C22⋊C411yescharacteristic
322Q8—28no—
323Q8—48no—
324Q8—44no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—28no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
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—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4#40812yes—
364C2×C4C2×#11712yes—
365C2×C4#52512yes—
366C2×C4#52511yescharacteristic
367C2×C4#52511yescharacteristic
368C2×C4C2×#13112yes—
369C2×C4#56812yes—
370C2×C4#57712yes—
371C2×C4#57812yes—
372C8—44no—
373C8—82no—
374C8—82no—
375C8—82no—
376C22—24no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22#228011yescharacteristic
382C22C2×#40811yescharacteristic
383C22#287911yescharacteristic
384C22#289812yes—
385C22#290112yes—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—44no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4#292414yes—
399C4#312112yes—
400C4#312611yescharacteristic
401C4#312711yescharacteristic
402C2—42no—
403C2#1746912yes—
404C2#1750612yes—
405C2#2071711yescharacteristic
406C2#2074511yescharacteristic
407C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8570C2semidirect
6. #8568C2semidirect
8. #8182C2semidirect
9. #8180C2semidirect

Maximal subgroups

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

subgroupindex
#340592
#219222
#214312
#87082
#85702
#85682
#83112
#81822
#81802