Finite Group Databasebeta

← #415541

Subgroups — #415541

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1846C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#14282C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11325C212yesmaximal
9#11324C212yesmaximal
10#11323C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#106C2212yes—
18C22×#106C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2212yes—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2212yes—
26C2×#376C2211yescharacteristic
27#1993—24no—
28#1990—24no—
29#1847—24no—
30#1846—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1446—22no—
38#1445—22no—
39#1444—22no—
40#1444—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—24no—
68C2×#107—42no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#107—42no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#106—42no—
77C2×#106—42no—
78C2×#106—42no—
79C2×#97—22no—
80C2×#97—22no—
81C2×#97D414yes—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8D414yes—
85#384—22no—
86#384—22no—
87#384—42no—
88#384—42no—
89#382—22no—
90#382—22no—
91#382—22no—
92#382—22no—
93#382—42no—
94#382—42no—
95#376—22no—
96#376—22no—
97C2×#95—24no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95D412yes—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336Q818yescomplemented
111#336Q818yescomplemented
112C22×Q8—24no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—82no—
116C22×Q8—82no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C2×D414yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D414yes—
140C22×C8C2×Q814yes—
141C22×C8C2×Q814yes—
142C22×C8C2×Q814yes—
143C22×C8C2×Q814yes—
144C4⋊Q8—28no—
145C4⋊Q8—28no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4—84no—
185C2×C4⋊C4C2×Q814yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×Q814yes—
193C2×C42C2×Q814yes—
194C2×C42C2×Q812yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#107—48no—
199#107—44no—
200#107—44no—
201#106—44no—
202#106—44no—
203#106—48no—
204#97—28no—
205C4×C8—28no—
206#95—24no—
207#95—24no—
208#95—48no—
209#95—42no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—84no—
214#95—84no—
215C24C22×D411yescharacteristic
216C2×Q8—48no—
217C2×Q8—88no—
218C2×Q8—88no—
219C2×Q8—84no—
220C2×Q8—84no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22⋊Q812yes—
258C22×C4C22⋊Q812yes—
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12712yes—
262C22×C4#12712yes—
263C22×C4C4⋊Q818yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C22×Q812yes—
268C22×C4C22×Q812yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8C4⋊Q814yes—
289C2×C8C4⋊Q814yes—
290C2×C8C4⋊Q818yes—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—816no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42C22⋊Q814yes—
320C42C22⋊Q814yes—
321C42#12314yes—
322C42#12314yes—
323C23C2×C22⋊Q811yescharacteristic
324C23C2×#12311yescharacteristic
325C23C2×#12711yescharacteristic
326C23C2×C4⋊Q812yes—
327C23#53512yes—
328C23#57612yes—
329C23#57712yes—
330C23#57712yes—
331C23#57812yes—
332Q8—816no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4#39914yes—
375C2×C4#39914yes—
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×#12311yescharacteristic
379C2×C4C2×#12311yescharacteristic
380C2×C4C2×C4⋊Q811yescharacteristic
381C2×C4C2×C4⋊Q811yescharacteristic
382C2×C4C2×C4⋊Q812yes—
383C2×C4#53511yescharacteristic
384C2×C4#53512yes—
385C2×C4#53511yescharacteristic
386C2×C4#55412yes—
387C2×C4#55412yes—
388C2×C4#55412yes—
389C2×C4#55412yes—
390C2×C4#55412yes—
391C2×C4#55412yes—
392C2×C4Q8212yes—
393C2×C4Q8212yes—
394C8—24no—
395C8—24no—
396C8—28no—
397C22C2×#39911yescharacteristic
398C22#250212yes—
399C22#256712yes—
400C22#265711yescharacteristic
401C22#286811yescharacteristic
402C22#308112yes—
403C22#308212yes—
404C22#309612yes—
405C22#309712yes—
406C22#315411yescharacteristic
407C22#315411yescharacteristic
408C22#315511yescharacteristic
409C22#315611yescharacteristic
410C22#316314yes—
411C22#316414yes—
412C22#322414yes—
413C22#322514yes—
414C4—24no—
415C4—24no—
416C4—48no—
417C4—84no—
418C4—162no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4#250211yescharacteristic
424C4#250211yescharacteristic
425C4#250212yes—
426C4#256711yescharacteristic
427C4#256712yes—
428C4#256711yescharacteristic
429C2#1519911yescharacteristic
430C2#2373511yescharacteristic
431C2#2373611yescharacteristic
432C2#2795914yes—
433C2#2921714yes—
434C2#3027411yescharacteristic
435C2#3027511yescharacteristic
436C2#3331811yescharacteristic
437C2#3331911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
110. #336Q8semidirect
111. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
C2×#19902
C2×#18472
C2×#18462
#143542
#142822
#113252
#113242
#113232
#106942