Finite Group Databasebeta

← #197989

Subgroups — #197989

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C2×#2986C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9228C214yesmaximal, complemented
7#9220C214yesmaximal, complemented
8#8038C211yesmaximal, characteristic
9#7880C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498—21no—
15C2×#487—21no—
16C2×#483—22no—
17C2×#475C2214yes—
18#2986—24no—
19#2986—24no—
20C2×#446—21no—
21C2×#446C2212yes—
22C2×C8×Q8—21no—
23C2×C8×Q8C2212yes—
24C2×#433—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
29#2285C2211yescharacteristic
30#1988—21no—
31#1981—21no—
32#1981—21no—
33#1981—21no—
34#1956C2214yes—
35#1894—22no—
36#1803C2214yes—
37#1787—22no—
38#1549—24no—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2214yes—
44C23×Q8—21no—
45C22×M16—21no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533D414yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8C2312yes—
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8D412yes—
63C2×C4×Q8—21no—
64C2×C4×Q8C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C42C2311yescharacteristic
70#498—28no—
71#487—28no—
72#483—216no—
73#475—28no—
74#475C2×C4116yes—
75#446—28no—
76#446—24no—
77#446—24no—
78C8×Q8—28no—
79C8×Q8—24no—
80C8×Q8—24no—
81#433—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431C2×C414yes—
87#431D414yes—
88C2×#106—24no—
89#423—22no—
90#423—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8D412yes—
107C2×#97—22no—
108C2×C4×C8—22no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2312yes—
112#384C2312yes—
113#382—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118#340—24no—
119#332—24no—
120#332—24no—
121C22×Q8—21no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—42no—
125C22×Q8—41no—
126C22×Q8—82no—
127C22×Q8—82no—
128C22×Q8C2×D412yes—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—42no—
134C2×M16—82no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8C22⋊C414yes—
153C4⋊Q8C22⋊C414yes—
154C4⋊Q8C22×C414yes—
155#125—24no—
156#125—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8C22⋊C418yes—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8C22×C414yes—
166C4×Q8C2×D414yes—
167#117—24no—
168#117—24no—
169#117—82no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—41no—
189C2×C42—22no—
190C2×C42—41no—
191C2×C42C22×C412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42#4111yescharacteristic
197C2×C42#4112yes—
198C2×C42C2411yescharacteristic
199#106—216no—
200C4⋊C8—28no—
201C4⋊C8—28no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—42no—
210C4⋊C8C22×C414yes—
211C4⋊C8C2×D412yes—
212C4⋊C8C2×D412yes—
213#103—216no—
214#103—84no—
215#103—84no—
216#103—84no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220#97—28no—
221C4×C8—28no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226#95—41no—
227#95—41no—
228C24#12111yescharacteristic
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—88no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8C2×C22⋊C414yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—84no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C4×D412yes—
259C22×C4C4×D412yes—
260C22×C4#12012yes—
261C22×C4#12112yes—
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12312yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269M16—84no—
270C2×C8—24no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4C2×C22⋊C414yes—
302C22⋊C4—28no—
303C22⋊C4—82no—
304C42—24no—
305C42—24no—
306C42—82no—
307C42C2×C22⋊C412yes—
308C42C4×D412yes—
309C42C4×D412yes—
310C42#12012yes—
311C42#12112yes—
312C42#12312yes—
313C42C23×C411yescharacteristic
314C42C22×D411yescharacteristic
315C42C22×D411yescharacteristic
316C42C2×#4111yescharacteristic
317C23—22no—
318C23—22no—
319C23—21no—
320C23—21no—
321C23#38612yes—
322C23C2×#12111yescharacteristic
323Q8—28no—
324Q8—48no—
325Q8—42no—
326Q8—42no—
327Q8—48no—
328Q8—88no—
329Q8—82no—
330Q8—82no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#38612yes—
355C2×C4#38612yes—
356C2×C4#38612yes—
357C2×C4#47112yes—
358C2×C4#47312yes—
359C2×C4C22×C22⋊C411yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×#12011yescharacteristic
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×#12311yescharacteristic
365C2×C4#57712yes—
366C2×C4#57712yes—
367C2×C4#57712yes—
368C2×C4#57812yes—
369C8—44no—
370C8—42no—
371C8—42no—
372C8—42no—
373C8—42no—
374C8—82no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22C2×#38611yescharacteristic
381C22#304312yes—
382C22#306312yes—
383C22C2×#47111yescharacteristic
384C22C2×#47311yescharacteristic
385C4—21no—
386C4—21no—
387C4—22no—
388C4—22no—
389C4—44no—
390C4—44no—
391C4—84no—
392C4—162no—
393C4#190814yes—
394C4#289812yes—
395C4#294912yes—
396C4#301011yescharacteristic
397C4#301011yescharacteristic
398C4#305711yescharacteristic
399C4#305911yescharacteristic
400C2—42no—
401C2#1956512yes—
402C2#1956712yes—
403C2#2093611yescharacteristic
404C2#2097411yescharacteristic
405C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9228C2semidirect
7. #9220C2semidirect
13. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#29862
#218942
#218042
C2×#15492
#92282
#92202
#80382
#78802