Finite Group Databasebeta

← #197438

Subgroups — #197438

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic
2#35334C211yesmaximal, characteristic
3#17708C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic
5#9194C214yesmaximal, complemented
6#9192C214yesmaximal, complemented
7C2×#1286C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2925—24no—
11#2925—24no—
12#2921C2214yes—
13#2920—22no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16#2879—21no—
17#2873—21no—
18#2851—22no—
19#2628C2212yes—
20#2628C2212yes—
21#2581—21no—
22#2581C412yes—
23#2580—21no—
24#2580C412yes—
25#2564—22no—
26#2564C414yes—
27#2316—21no—
28#2311—21no—
29C2×#382C2211yescharacteristic
30#1550—24no—
31#1549C2214yes—
32#1548C2214yes—
33#1542—24no—
34#1461—21no—
35#1460—21no—
36#1459—22no—
37#1458C2214yes—
38#1286C2218yes—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C2×#117C2312yes—
65C22×C4⋊C4—21no—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#433—216no—
69#432—24no—
70#432—24no—
71#431—24no—
72#431—24no—
73#431—22no—
74#431—22no—
75#431—22no—
76#431C2314yes—
77#423—22no—
78#423—22no—
79#423D414yes—
80#423C2314yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8D414yes—
84#408—28no—
85#407—24no—
86C2×C22⋊C8—22no—
87#405—28no—
88C4×M16—24no—
89C2×C4×C8—22no—
90#400—22no—
91#400C2×C414yes—
92#399—22no—
93#399C2×C414yes—
94#395—22no—
95#395C2×C414yes—
96#393—22no—
97#393C2×C414yes—
98#391—22no—
99#390—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#386—24no—
107#385—24no—
108#385—24no—
109#382—21no—
110#382—21no—
111#382C2×C412yes—
112#382C2312yes—
113#380—22no—
114#380—22no—
115#380—24no—
116C2×#95—21no—
117C2×#95C2×C412yes—
118#333—28no—
119#332—28no—
120#324—24no—
121#323—24no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—24no—
125C2×#41—24no—
126C22×Q8—21no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C412yes—
129C22×D4C22⋊C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—24no—
143C22×C8—42no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148#125—41no—
149#125—41no—
150#125—41no—
151#125—41no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22×C412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8C2×D418yes—
221C22⋊C8—28no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225#97—28no—
226C4×C8—28no—
227C4×C8—24no—
228C4×C8—42no—
229#95—22no—
230#95—22no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95C22×C414yes—
237C24C2×C22⋊C412yes—
238C24C2×C22⋊C411yescharacteristic
239#41—44no—
240#41—44no—
241#41—48no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—42no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—44no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C4×D412yes—
280C22×C4C4×D412yes—
281C22×C4#12012yes—
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4#12312yes—
285C22×C4#13514yes—
286C22×C4#13714yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290M16—44no—
291M16—44no—
292M16—48no—
293C2×C8—28no—
294C2×C8—48no—
295C2×C8—44no—
296C2×C8—84no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—44no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—44no—
328C42C4×D412yes—
329C42C4×D412yes—
330C42#12011yescharacteristic
331C42#12011yescharacteristic
332C42#12112yes—
333C42#12112yes—
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42C22×D412yes—
337C42C2×#4112yes—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23#38612yes—
346C23#41612yes—
347C23#41912yes—
348C23C22×C22⋊C411yescharacteristic
349Q8—44no—
350Q8—84no—
351Q8—84no—
352D4—48no—
353D4—44no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—164no—
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38611yescharacteristic
388C2×C4#38611yescharacteristic
389C2×C4#38612yes—
390C2×C4#41612yes—
391C2×C4#41912yes—
392C2×C4#45214yes—
393C2×C4#45214yes—
394C2×C4#48518yes—
395C2×C4C2×C4×D412yes—
396C2×C4C2×#12011yescharacteristic
397C2×C4C2×#12112yes—
398C2×C4C2×#12311yescharacteristic
399C2×C4C2×#13512yes—
400C2×C4C2×#13712yes—
401C8—48no—
402C8—84no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22#187314yes—
415C22C2×#38611yescharacteristic
416C22C2×#41611yescharacteristic
417C22C2×#41911yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—84no—
428C4—82no—
429C4—82no—
430C4#187314yes—
431C4C2×#45212yes—
432C4C2×#48512yes—
433C2—21no—
434C2—21no—
435C2—24no—
436C2—82no—
437C2C2×#187311yescharacteristic
438C2#2100911yescharacteristic
439C2#2101711yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
3. #17708C2semidirect
5. #9194C2semidirect
6. #9192C2semidirect
7. C2×#1286C2semidirect

Maximal subgroups

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

subgroupindex
#601252
#353342
#177082
#173492
#91942
#91922
C2×#12862