Finite Group Databasebeta

← #201164

Subgroups — #201164

Order 512.

1067 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#17676C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9450C218yesmaximal, complemented
6#7952C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2983—24no—
10#2983—22no—
11#2983C2214yes—
12C2×#446—24no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C2×#431—21no—
17#2855—21no—
18#2855—21no—
19#2627—21no—
20#2627—21no—
21#2582—22no—
22#2581—22no—
23#2568C412yes—
24#2567C412yes—
25#2565C2214yes—
26#2317C412yes—
27#2310C412yes—
28#2285C2211yescharacteristic
29#1534C2218yes—
30#1531—24no—
31#1531—24no—
32#1467—22no—
33#1467C2214yes—
34C2×#324—21no—
35C2×#324—21no—
36#1285C2218yes—
37C23×Q8C2×C411yescharacteristic
38#533—22no—
39#533—22no—
40#533—22no—
41#533—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125C2312yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8C2312yes—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42C2311yescharacteristic
65#446—216no—
66#446—216no—
67#432—24no—
68#432—24no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432D414yes—
75#432C2314yes—
76#432C2314yes—
77#431—24no—
78#431—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97D414yes—
88C2×C4×C8—22no—
89#400—22no—
90#400—22no—
91#398—24no—
92#398—24no—
93#395—22no—
94#395—22no—
95#391—24no—
96#391—22no—
97#391—22no—
98#391C2×C414yes—
99#391C2×C414yes—
100#389—24no—
101#389—22no—
102#389—22no—
103#387—24no—
104#387—24no—
105#385—22no—
106#385—22no—
107#384—21no—
108#384C2×C414yes—
109#382—21no—
110#382C2×C414yes—
111#381—22no—
112#381—22no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2312yes—
117#330—216no—
118#324—24no—
119#324—24no—
120C22×Q8—24no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—42no—
126C22×Q8C22⋊C412yes—
127C22×Q8C22×C412yes—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C22×C8—24no—
136C22×C8—42no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141#125—24no—
142#125—24no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—42no—
146C22⋊Q8—42no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—24no—
162#117—24no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C412yes—
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×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C414yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221#97—28no—
222#97—24no—
223#97—24no—
224#97—42no—
225#97C2×D418yes—
226C4×C8—28no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—82no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4#10114yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—28no—
289C2×C8—48no—
290C2×C8—44no—
291C2×C8—84no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—44no—
324C42C4×D412yes—
325C42C4×D412yes—
326C42#12012yes—
327C42#12112yes—
328C42#12112yes—
329C42#12312yes—
330C42C22×D412yes—
331C42C2×#4112yes—
332C23—41no—
333C23—41no—
334C23#38611yescharacteristic
335C23#38611yescharacteristic
336C23C2×#10112yes—
337C23#41312yes—
338C23C22×C22⋊C411yescharacteristic
339Q8—48no—
340Q8—48no—
341Q8—82no—
342Q8—84no—
343Q8—82no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4#38611yescharacteristic
385C2×C4#38612yes—
386C2×C4#38612yes—
387C2×C4#38611yescharacteristic
388C2×C4C2×#10114yes—
389C2×C4C2×C4×D412yes—
390C2×C4C2×#12011yescharacteristic
391C2×C4C2×#12112yes—
392C2×C4C2×#12311yescharacteristic
393C8—48no—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22#190112yes—
404C22#190112yes—
405C22C2×#38611yescharacteristic
406C22C22×#10111yescharacteristic
407C22C2×#41311yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—84no—
420C4—82no—
421C4—82no—
422C4#191212yes—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2#2094711yescharacteristic
428C2C2×#190111yescharacteristic
429C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9450C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351792
#176762
C2×#14672
#94502
#79522