Finite Group Databasebeta

← #115869

Subgroups — #115869

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21312C211yesmaximal, characteristic
2C2×#1518C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#13954C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#389—22no—
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1989—22no—
17#1988C412yescomplemented
18#1987C412yescomplemented
19#1736—22no—
20#1518—22no—
21#1518—22no—
22#1503—22no—
23#1503—22no—
24C2×#328—22no—
25C2×#328—22no—
26#1444—22no—
27C4⋊#95—22no—
28#1303—22no—
29#1286—22no—
30#1279—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533D412yes—
39#533Q812yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431D412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#389—44no—
76#389—44no—
77#382—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#328—28no—
97#328—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—42no—
109C2×M16—42no—
110C22×C8—22no—
111C22×C8—41no—
112C22×C8—41no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C4⋊C412yescomplemented
120C4⋊Q8C4⋊C412yescomplemented
121#125—22no—
122#125—22no—
123#125—41no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—22no—
142#117—22no—
143#117—41no—
144#117—41no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4SD1614yes—
166C2×C4⋊C4Q1614yes—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—82no—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8—41no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—48no—
211#95—41no—
212#95—41no—
213C24#12011yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—81no—
219C2×Q8—81no—
220C2×Q8—82no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12711yescharacteristic
261C22×C4C2×SD1612yes—
262C22×C4C2×Q1612yes—
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265M16—82no—
266M16—81no—
267M16—81no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—81no—
273C2×C8—81no—
274C2×C8—82no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—816no—
293C4⋊C4—82no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4#10318yes—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—42no—
300C42—42no—
301C42—82no—
302C42—82no—
303C42—82no—
304C42—81no—
305C42—81no—
306C42C2×C4⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12611yescharacteristic
321C42#12611yescharacteristic
322C23—21no—
323C23—21no—
324C23—22no—
325C23#38611yescharacteristic
326C23#39211yescharacteristic
327C23#39311yescharacteristic
328C23#45012yes—
329C23#45112yes—
330Q8—82no—
331Q8—82no—
332Q8—82no—
333Q8—82no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4#38011yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#39011yescharacteristic
376C2×C4#39411yescharacteristic
377C2×C4#39411yescharacteristic
378C2×C4#39611yescharacteristic
379C2×C4#39611yescharacteristic
380C2×C4#39711yescharacteristic
381C2×C4#39911yescharacteristic
382C2×C4C2×#10312yes—
383C2×C4#41712yes—
384C2×C4#46012yes—
385C2×C4#46212yes—
386C2×C4#47512yes—
387C2×C4#47712yes—
388C2×C4#48312yes—
389C2×C4#48412yes—
390C2×C4#48712yes—
391C2×C4#48912yes—
392C2×C4#49211yescharacteristic
393C2×C4#49411yescharacteristic
394C2×C4#49611yescharacteristic
395C2×C4#49711yescharacteristic
396C8—82no—
397C8—82no—
398C8—82no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#143911yescharacteristic
408C22#187212yes—
409C22#190411yescharacteristic
410C22#190511yescharacteristic
411C22#190612yes—
412C22#201411yescharacteristic
413C22#201611yescharacteristic
414C22#202412yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—48no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—162no—
428C4#180312yes—
429C4#195611yescharacteristic
430C4#195911yescharacteristic
431C4#207212yes—
432C4#209511yescharacteristic
433C4#209611yescharacteristic
434C2—21no—
435C2—21no—
436C2—22no—
437C2#917512yes—
438C2#917612yes—
439C2#1105011yescharacteristic
440C2#1105711yescharacteristic
441C2#1106011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1988C4semidirect
18. #1987C4semidirect
119. C4⋊Q8C4⋊C4semidirect
120. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#213122
C2×#15182
C2×#15032
#139542
C2×#13032
C2×#12862
#78922