Finite Group Databasebeta

← #121843

Subgroups — #121843

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#8115C211yesmaximal, characteristic
3C2×#1305C211yesmaximal, characteristic
4#8019C211yesmaximal, characteristic
5#7983C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7929C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811C414yescomplemented
18#1306—22no—
19#1305—22no—
20#1294—22no—
21#1289—22no—
22#1284—22no—
23#1282—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16C2×C412yes—
29C2×#117—21no—
30C22×C4⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C42—21no—
34C2×#108—24no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407C2×C412yes—
49#407C2×C412yes—
50#407Q812yes—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D412yes—
58C2×C22⋊C8D412yes—
59#381D412yes—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65#380D412yes—
66#379—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4Q812yes—
73C2×#95—21no—
74C2×#95—21no—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—82no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—82no—
103C2×M16—82no—
104C2×M16—82no—
105C2×M16—82no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8C22⋊C414yes—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—82no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—82no—
124C2×C42—22no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—82no—
128#108—84no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—41no—
150#95—41no—
151C24—42no—
152C24—41no—
153C24—41no—
154C24—81no—
155C24C4×Q811yescharacteristic
156C24C22⋊Q811yescharacteristic
157C24#12311yescharacteristic
158C24#12311yescharacteristic
159C24#12411yescharacteristic
160C24#12611yescharacteristic
161C24#12611yescharacteristic
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4#11711yescharacteristic
188C22×C4#11711yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4C4×Q811yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12411yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12511yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4#12611yescharacteristic
212C22×C4#12711yescharacteristic
213C22×C4C4⋊Q811yescharacteristic
214M16—44no—
215M16—44no—
216M16—82no—
217M16—82no—
218M16—82no—
219M16—82no—
220M16—82no—
221M16—82no—
222C2×C8—24no—
223C2×C8—44no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C22⋊C4—24no—
234C22⋊C4—42no—
235C22⋊C4—44no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—82no—
243C42—82no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—84no—
270C23—84no—
271C23#38711yescharacteristic
272C23#38711yescharacteristic
273C23#39111yescharacteristic
274C23#39711yescharacteristic
275C23#39711yescharacteristic
276C23#39911yescharacteristic
277C23#40011yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—164no—
318C2×C4#38111yescharacteristic
319C2×C4#38711yescharacteristic
320C2×C4#38711yescharacteristic
321C2×C4#39011yescharacteristic
322C2×C4#39811yescharacteristic
323C2×C4#39811yescharacteristic
324C2×C4#39911yescharacteristic
325C2×C4#40011yescharacteristic
326C8—44no—
327C8—82no—
328C8—82no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22#144511yescharacteristic
360C22#208111yescharacteristic
361C22#208211yescharacteristic
362C22#210212yes—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—84no—
371C4—162no—
372C4—162no—
373C4#182812yes—
374C2—21no—
375C2—21no—
376C2—22no—
377C2—41no—
378C2—41no—
379C2—41no—
380C2—41no—
381C2—41no—
382C2#1135111yescharacteristic
383C2#1136211yescharacteristic
384C2#1136711yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1811C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81152
C2×#13052
#80192
#79832
#79422
#79292