Finite Group Databasebeta

← #201195

Subgroups — #201195

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2#34955C211yesmaximal, characteristic
3#17731C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9458C214yesmaximal, complemented
6#9456C214yesmaximal, complemented
7#7949C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983—22no—
12#2983C2214yes—
13C2×#446—24no—
14#2920C2212yes—
15C2×#432—21no—
16C2×#432C2212yes—
17C2×#431—21no—
18#2592—22no—
19#2591—21no—
20#2591—21no—
21#2568C2212yes—
22#2548C2212yes—
23#2504C412yes—
24#2436C412yes—
25#2398—22no—
26C4×#124C414yes—
27#2297—21no—
28#2297—21no—
29C4×#117C2211yescharacteristic
30#1557—24no—
31#1556C2214yes—
32#1555C2214yes—
33#1540—24no—
34#1467C2214yes—
35#1463—21no—
36#1463—21no—
37#1461—22no—
38#1285C2218yes—
39C22×#41—21no—
40C23×Q8C2×C411yescharacteristic
41#532—22no—
42#532—22no—
43#532—22no—
44#532—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2311yescharacteristic
47C2×#124—21no—
48C2×#124C2×C414yes—
49C2×#121—21no—
50C2×#121C2×C412yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8C2312yes—
59C2×C4×D4—21no—
60C2×C4×D4C2×C412yes—
61C2×#117—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#446—216no—
69#446—216no—
70#432—24no—
71#432—24no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432D414yes—
76#432C2314yes—
77#432C2314yes—
78#431—24no—
79#431—24no—
80#431—22no—
81#431—22no—
82#423—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8D414yes—
86#407—22no—
87C2×C22⋊C8—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×C4×C8—22no—
92#400—22no—
93#400—22no—
94#397—24no—
95#394—22no—
96#394—22no—
97#393—24no—
98#391—22no—
99#391—22no—
100#391C2×C414yes—
101#390—22no—
102#390—22no—
103#390C2×C414yes—
104#388—24no—
105#388—24no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#382C2×C414yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4C2×C412yes—
117#376—21no—
118C43C2×C412yes—
119#332—28no—
120#331—28no—
121#324—24no—
122#324—24no—
123C2×#41—48no—
124C2×#41—41no—
125C2×#41—41no—
126C2×#41—41no—
127C2×#41—41no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—44no—
132C22×Q8C22⋊C412yes—
133C22×Q8C22×C412yes—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C412yes—
138C23×C4—22no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—42no—
147C22×C8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150#127—22no—
151#127—22no—
152#124—48no—
153#124—42no—
154#124—42no—
155#121—24no—
156#121—24no—
157#121—42no—
158#121—42no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—42no—
170C4×D4—42no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C414yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—42no—
222C4⋊C8C2×D418yes—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—42no—
227#97—28no—
228#97—24no—
229#97—24no—
230#97—24no—
231C4×C8—28no—
232C4×C8—42no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239C24—22no—
240C24C2×C22⋊C411yescharacteristic
241#41—84no—
242#41—84no—
243#41—84no—
244#41—84no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—44no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—24no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—48no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—82no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12311yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304M16—84no—
305C2×C8—28no—
306C2×C8—48no—
307C2×C8—44no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—48no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—44no—
334C42—82no—
335C42C4×D412yes—
336C42C4×D412yes—
337C42#12012yes—
338C42#12112yes—
339C42#12112yes—
340C42#12312yes—
341C42C22×D412yes—
342C42C2×#4112yes—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23#38611yescharacteristic
351C23#38611yescharacteristic
352C23C22×C22⋊C411yescharacteristic
353Q8—48no—
354Q8—48no—
355Q8—82no—
356Q8—82no—
357D4—84no—
358D4—82no—
359D4—84no—
360D4—84no—
361D4—84no—
362D4—82no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#38611yescharacteristic
404C2×C4#41314yes—
405C2×C4#45414yes—
406C2×C4C2×C4×D412yes—
407C2×C4C2×#12011yescharacteristic
408C2×C4C2×#12112yes—
409C2×C4C2×#12311yescharacteristic
410C8—48no—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—44no—
420C22—88no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22#188812yes—
425C22C2×#38611yescharacteristic
426C22C2×#41312yes—
427C4—22no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—24no—
432C4—22no—
433C4—22no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—84no—
438C4—82no—
439C4#188812yes—
440C4C2×#45412yes—
441C2—22no—
442C2—22no—
443C2—41no—
444C2—162no—
445C2C2×#188811yescharacteristic
446C2#2098911yescharacteristic
447C2#2100911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
4. #17365C2semidirect
5. #9458C2semidirect
6. #9456C2semidirect
7. #7949C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#349552
#177312
#173652
#94582
#94562
#79492