Finite Group Databasebeta

← #197385

Subgroups — #197385

Order 512.

1077 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17672C212yesmaximal
5#9190C218yesmaximal, complemented
6#7977C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2983—24no—
10#2983—22no—
11#2983C2214yes—
12C2×#446—24no—
13#2920C2212yes—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#431—21no—
17#2855—21no—
18#2855—21no—
19#2634C2214yes—
20#2581—22no—
21#2568C412yes—
22#2546C412yes—
23#2503—21no—
24#2484C412yes—
25#2484C412yes—
26#2434—21no—
27#2327—22no—
28#2283C2211yescharacteristic
29#1821—21no—
30#1801—22no—
31#1784—21no—
32C4×#101C2214yes—
33#1537—24no—
34#1531—24no—
35#1528C2218yes—
36#1288C2218yes—
37C23×Q8D411yescharacteristic
38#533—22no—
39#533—22no—
40#533—22no—
41#533—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125C2312yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8C2312yes—
58C2×#117C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
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#432D414yes—
74#432C2314yes—
75#431—24no—
76#431—24no—
77#431—22no—
78#431—22no—
79#423—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#101—24no—
84#407—22no—
85C2×C22⋊C8—22no—
86C4×M16C2314yes—
87C2×#97—22no—
88C2×#97D414yes—
89C2×C4×C8—22no—
90#400—24no—
91#400C2×C414yes—
92#400C2×C414yes—
93#398C2×C414yes—
94#398C2×C414yes—
95#395—24no—
96#391—22no—
97#391—22no—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#385—22no—
107#385—22no—
108#383C2312yes—
109#382—22no—
110#382—21no—
111#381—22no—
112#381—22no—
113#381C2×C412yes—
114#380—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C22⋊C4—21no—
117#376—21no—
118#343—24no—
119#330—216no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—44no—
125C22×Q8C22⋊C412yes—
126C22×Q8C2×D412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×M16—24no—
135C22×C8—24no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—48no—
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—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214#101—48no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219#97—28no—
220#97—24no—
221#97—24no—
222#97C2×D418yes—
223C4×C8—28no—
224C4×C8—24no—
225#95—22no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95C22×C414yes—
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—44no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C4×D412yes—
275C22×C4C4×D412yes—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12112yes—
279C22×C4#12112yes—
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—48no—
286C2×C8—28no—
287C2×C8—48no—
288C2×C8—44no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—48no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—48no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—44no—
318C42—44no—
319C42C4×D412yes—
320C42C4×D412yes—
321C42#12012yes—
322C42#12112yes—
323C42#12112yes—
324C42#12312yes—
325C42C22×D412yes—
326C42C2×#4112yes—
327C23—22no—
328C23—22no—
329C23—41no—
330C23#38611yescharacteristic
331C23#38611yescharacteristic
332C23C22×C22⋊C411yescharacteristic
333Q8—48no—
334Q8—48no—
335Q8—82no—
336Q8—84no—
337Q8—82no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#38611yescharacteristic
377C2×C4#38612yes—
378C2×C4#38612yes—
379C2×C4#38611yescharacteristic
380C2×C4#42114yes—
381C2×C4C2×C4×D412yes—
382C2×C4C2×#12011yescharacteristic
383C2×C4C2×#12112yes—
384C2×C4C2×#12311yescharacteristic
385C8—48no—
386C8—84no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22#191512yes—
396C22C2×#38611yescharacteristic
397C22C2×#42112yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—24no—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—84no—
410C4—82no—
411C4—82no—
412C4#191212yes—
413C2—22no—
414C2—22no—
415C2—41no—
416C2#2090211yescharacteristic
417C2C2×#191211yescharacteristic
418C2C2×#191511yescharacteristic
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. #9190C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#352482
#207272
#176722
#91902
#79772