Finite Group Databasebeta

← #197391

Subgroups — #197391

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2634C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17667C212yesmaximal
5#9190C218yesmaximal, complemented
6#7975C212yesmaximal
7C22×#125C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—21no—
11#2983C2214yes—
12#2982—21no—
13Q8×M16—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2634—22no—
18#2634—21no—
19#2634—21no—
20#2634C412yes—
21#2634C412yes—
22#2634C2214yes—
23C2×#400—21no—
24C2×#400C412yes—
25C2×#398C412yes—
26C2×#395—21no—
27C2×#387—22no—
28C2×#383C2211yescharacteristic
29#1821—21no—
30#1801—22no—
31#1784—21no—
32C4×#101C2214yes—
33#1534—24no—
34#1528—24no—
35#1528C2218yes—
36#1288C2218yes—
37C23×Q8D411yescharacteristic
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×#125C2312yes—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2312yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#446—28no—
66#446—216no—
67C8×Q8—28no—
68#432—24no—
69#432—24no—
70#432—22no—
71#432—22no—
72#432D414yes—
73#432C2314yes—
74#431—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—24no—
81#423—22no—
82#423—22no—
83C2×#101—24no—
84#407—22no—
85#407—22no—
86C4×M16—24no—
87C4×M16D414yes—
88C4×M16C2314yes—
89#400—22no—
90#400—22no—
91#400—22no—
92#400—22no—
93#400—24no—
94#400C2×C414yes—
95#400C2×C414yes—
96#398—22no—
97#398—22no—
98#398C2×C414yes—
99#398C2×C414yes—
100#395—22no—
101#395—22no—
102#395—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#383—21no—
108#383—21no—
109#383C2×C412yes—
110#383C2312yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2×C412yes—
117#343—24no—
118#330—216no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C2×D412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135#125—24no—
136#125—24no—
137#125—24no—
138#125—24no—
139#125—42no—
140#125—42no—
141#125—42no—
142#125—42no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—24no—
154C4×Q8—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—28no—
204C4⋊C8—28no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209#101—48no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214#97—28no—
215#97—24no—
216#97C2×D418yes—
217C4×C8—28no—
218C4×C8—24no—
219C4×C8—24no—
220#95—22no—
221#95—22no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95C22×C414yes—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—48no—
241C2×Q8—48no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12311yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—48no—
283M16—48no—
284C2×C8—28no—
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42C4×D412yes—
314C42C4×D412yes—
315C42#12012yes—
316C42#12112yes—
317C42#12112yes—
318C42#12312yes—
319C42C22×D412yes—
320C42C2×#4112yes—
321C23—22no—
322C23—22no—
323C23—22no—
324C23#38611yescharacteristic
325C23#38611yescharacteristic
326C23C22×C22⋊C411yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—88no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#38611yescharacteristic
370C2×C4#38611yescharacteristic
371C2×C4#38612yes—
372C2×C4#38612yes—
373C2×C4#42114yes—
374C2×C4C2×C4×D412yes—
375C2×C4C2×#12011yescharacteristic
376C2×C4C2×#12112yes—
377C2×C4C2×#12311yescharacteristic
378C8—48no—
379C8—84no—
380C22—28no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22#191212yes—
388C22#191512yes—
389C22C2×#38611yescharacteristic
390C22C2×#42112yes—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—84no—
404C4—82no—
405C4—82no—
406C2—22no—
407C2—22no—
408C2—22no—
409C2#2090211yescharacteristic
410C2C2×#191211yescharacteristic
411C2C2×#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
#602482
C2×#26342
#207272
#176672
#91902
#79752