Finite Group Databasebeta

← #127621

Subgroups — #127621

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1800C211yesmaximal, characteristic
2C2×#1459C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#423—21no—
7C22×#100—21no—
8C22×C22⋊C8—21no—
9C2×#386—21no—
10C2×#380C411yescharacteristic
11C22×#95C411yescharacteristic
12#1800—22no—
13#1800C414yescomplemented
14C2×#343C2212yes—
15#1459—22no—
16#1459C414yescomplemented
17C2×#324C2212yes—
18C2×#323—21no—
19#1441C2212yes—
20#1439C412yes—
21#1434—21no—
22C22⋊#95—21no—
23C22⋊#95—21no—
24C23×D4D411yescharacteristic
25C24×C4—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C22×M16—22no—
29C23×C8—41no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
32C2×C4×D4—42no—
33C2×C4×D4C2×C412yes—
34C2×C4×D4C2×C412yes—
35C2×C4×D4D412yes—
36C2×C4×D4Q812yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42D411yescharacteristic
44#423—24no—
45#423—42no—
46C2×C4⋊C8—41no—
47C2×C4⋊C8—41no—
48C2×#100—24no—
49C2×#100—42no—
50C2×#100—42no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—41no—
55#386—42no—
56#386—42no—
57#380—41no—
58#380—41no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—42no—
65C2×#95C2×C412yes—
66#343—22no—
67#343C2×C414yescomplemented
68#324—22no—
69#324C2×C414yescomplemented
70#323—24no—
71C25C22⋊C411yescharacteristic
72C25C22⋊C411yescharacteristic
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4C22⋊C412yes—
78C22×D4C4⋊C412yes—
79C22×D4D812yes—
80C22×D4SD1612yes—
81C22×D4SD1612yes—
82C22×D4Q1612yes—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—44no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—82no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C4212yes—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C22×C8—42no—
114C22×C8—44no—
115C22×C8—42no—
116C22×C8—84no—
117C4×D4—42no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4C22×C411yescharacteristic
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×Q811yescharacteristic
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—82no—
143C2×C22⋊C4C4212yes—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C4⋊C412yes—
146C2×C22⋊C4C4⋊C412yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42C22⋊C412yes—
151C2×C42C4⋊C412yes—
152C4⋊C8—44no—
153C4⋊C8—44no—
154#100—48no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C22⋊C8—44no—
158#95—42no—
159#95—44no—
160#95—42no—
161#95—41no—
162#95—41no—
163#95—42no—
164#95—42no—
165#95—82no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24#9512yes—
179C24#9512yes—
180C24#10212yes—
181C24#10212yes—
182C24#10312yes—
183C24#10312yes—
184C24C2×C22⋊C411yescharacteristic
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4#10214yes—
198C2×D4#10314yes—
199C2×D4#10614yes—
200C2×D4#10714yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—48no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—88no—
249C22×C4—88no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4#9512yes—
253C22×C4#9512yes—
254C22×C4#9911yescharacteristic
255C22×C4#9911yescharacteristic
256C22×C4#9912yes—
257C22×C4C2×C4211yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×SD1611yescharacteristic
265C22×C4C2×SD1611yescharacteristic
266C22×C4C2×Q1611yescharacteristic
267M16—44no—
268M16—44no—
269M16—82no—
270M16—82no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—88no—
276C2×C8—84no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—84no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—88no—
287C22⋊C4—84no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—42no—
307C23—44no—
308C23—42no—
309C23—44no—
310C23—42no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—82no—
325C23—82no—
326C23#34014yes—
327C23#34014yes—
328C23C2×#9511yescharacteristic
329C23C2×#9911yescharacteristic
330C23C2×#9911yescharacteristic
331C23C2×#10211yescharacteristic
332C23C2×#10311yescharacteristic
333C23#42111yescharacteristic
334C23#42111yescharacteristic
335D4—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—816no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#34212yes—
395C2×C4#34212yes—
396C2×C4C2×#10211yescharacteristic
397C2×C4C2×#10311yescharacteristic
398C2×C4C2×#10611yescharacteristic
399C2×C4C2×#10711yescharacteristic
400C8—84no—
401C8—84no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—24no—
411C22—24no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—44no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22#151911yescharacteristic
429C22#151911yescharacteristic
430C22#152011yescharacteristic
431C22#152011yescharacteristic
432C22#173311yescharacteristic
433C22C2×#34011yescharacteristic
434C22C2×#34211yescharacteristic
435C22#212112yes—
436C22#213212yes—
437C4—22no—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—42no—
448C4—84no—
449C4—162no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—44no—
455C2—44no—
456C2#802311yescharacteristic
457C2#1188412yes—
458C2#1190612yes—
459C2#1257411yescharacteristic
460C2#1258511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
13. #1800C4semidirect
16. #1459C4semidirect
67. #343C2×C4semidirect
69. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×#14592
#139602
#72512