Finite Group Databasebeta

← #137714

Subgroups — #137714

Order 512.

678 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
1C22×C22⋊C16C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3#12679C211yesmaximal, characteristic
4C2×#1322C212yesmaximal, complemented
5#7183C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C23×C16—21no—
8C22×C4⋊C8C411yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C22×C22⋊C8C411yescharacteristic
11C2×C4×C22⋊C4C411yescharacteristic
12C2×C22⋊C16—24no—
13C2×C22⋊C16—21no—
14C2×C22⋊C16—21no—
15C2×C22⋊C16—21no—
16C2×C22⋊C16C2212yes—
17C2×C22⋊C16C2212yes—
18#1850—21no—
19#1850—21no—
20#1850—21no—
21#1850—21no—
22C2×#336C411yescharacteristic
23C2×#336C2211yescharacteristic
24#1322—24no—
25C24×C4C2×C411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C23×C8—42no—
29C23×C8—42no—
30C23×C8C2×C411yescharacteristic
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C23×C8D411yescharacteristic
37C23×C8C2311yescharacteristic
38C22×C22⋊C4C811yescharacteristic
39C22×C22⋊C4C811yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C16—24no—
43C22×C16—22no—
44C22×C16—22no—
45C22×C16—42no—
46C22×C16—42no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—42no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—42no—
57C4×C22⋊C4—21no—
58C4×C22⋊C4—21no—
59C4×C22⋊C4—21no—
60C4×C22⋊C4—21no—
61C2×#95C811yescharacteristic
62C2×#95C811yescharacteristic
63C22⋊C16—24no—
64C22⋊C16—22no—
65C22⋊C16—22no—
66C22⋊C16—22no—
67C22⋊C16—42no—
68C22⋊C16—42no—
69C22⋊C16D414yes—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74C25C2×C811yescharacteristic
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C2×C811yescharacteristic
84C23×C4C2×C811yescharacteristic
85C23×C4C2×C811yescharacteristic
86C23×C4C2×C811yescharacteristic
87C23×C4C2×C811yescharacteristic
88C23×C4M1611yescharacteristic
89C23×C4M1611yescharacteristic
90C23×C4C22×C411yescharacteristic
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—48no—
115C22×C8—48no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8C22⋊C411yescharacteristic
121C22×C8C22⋊C411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D412yes—
125C22×C8#4112yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4C1614yescomplemented
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C16—24no—
142C2×C16—48no—
143C2×C16—42no—
144C2×C16—42no—
145C2×C16—42no—
146C2×C16—42no—
147C2×C16—42no—
148C4⋊C8—42no—
149C4⋊C8—42no—
150C4⋊C8—41no—
151C4⋊C8—42no—
152C4⋊C8—41no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—41no—
164C24—41no—
165C24C22⋊C811yescharacteristic
166C24C22⋊C811yescharacteristic
167C24C2×C1612yes—
168C24M3212yes—
169C24C22×C811yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4C22⋊C811yescharacteristic
214C22×C4C22⋊C811yescharacteristic
215C22×C4#9912yes—
216C22×C4#10112yes—
217C22×C4C2×C1612yescomplemented
218C22×C4C2×C1612yescomplemented
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C4×D412yes—
221C22×C4C2×M1611yescharacteristic
222C22×C4#13112yes—
223C2×C8—24no—
224C2×C8—24no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8#9912yes—
250C2×C8#12011yescharacteristic
251C2×C8#12112yes—
252C2×C8#12311yescharacteristic
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—42no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—42no—
263C16—44no—
264C16—82no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—24no—
274C23—24no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—24no—
280C23—24no—
281C23—21no—
282C23—21no—
283C23—42no—
284C23—42no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23#32412yes—
290C23C22⋊C1612yes—
291C23C22⋊C1612yes—
292C23#34912yes—
293C23C2×C22⋊C811yescharacteristic
294C23C8×D412yes—
295C23C22×C1611yescharacteristic
296C23C2×M3211yescharacteristic
297C23#50412yes—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4#32412yes—
339C2×C4#38611yescharacteristic
340C2×C4#40811yescharacteristic
341C2×C4C2×#9911yescharacteristic
342C2×C4C2×#9911yescharacteristic
343C2×C4C2×#10111yescharacteristic
344C2×C4#41311yescharacteristic
345C2×C4#43512yes—
346C8—21no—
347C8—22no—
348C8—22no—
349C8—21no—
350C8—41no—
351C8—81no—
352C8—81no—
353C22—22no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—48no—
371C22—48no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#132214yescomplemented
377C22C2×#32411yescharacteristic
378C22#145911yescharacteristic
379C22#186911yescharacteristic
380C22C2×C22⋊C1611yescharacteristic
381C22#211311yescharacteristic
382C22C2×#34911yescharacteristic
383C22#211511yescharacteristic
384C22D4×C1612yes—
385C22#216812yes—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4#146811yescharacteristic
394C4#147211yescharacteristic
395C4#189811yescharacteristic
396C4#190111yescharacteristic
397C2—24no—
398C2—22no—
399C2—22no—
400C2—42no—
401C2#908111yescharacteristic
402C2C2×#132211yescharacteristic
403C2#1144111yescharacteristic
404C2#1146511yescharacteristic
405C2#1146711yescharacteristic
406C2#1297611yescharacteristic
407C2#1298511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1322C2semidirect
5. #7183C2semidirect
132. C2×C22⋊C4C16semidirect
217. C22×C4C2×C16semidirect
218. C22×C4C2×C16semidirect
376. C22#1322semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C162
C2×#18502
#126792
C2×#13222
#71832