Finite Group Databasebeta

← #181545

Subgroups — #181545

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3C2×#2353C211yesmaximal, characteristic
4#8024C212yesmaximal, complemented
5#8022C212yesmaximal, complemented
6#8020C212yesmaximal, complemented
7#8018C212yesmaximal, complemented
8#8016C212yesmaximal, complemented
9#8012C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2973—24no—
12#2973C418yescomplemented
13C2×#435—24no—
14C2×#432C2212yes—
15C2×#407C2212yes—
16C22×C22⋊C8C2212yes—
17#2357—24no—
18#2354C418yescomplemented
19#2353—24no—
20#2353C418yescomplemented
21#2312—21no—
22#2312—21no—
23#2309—21no—
24C2×#388—21no—
25C2×#387—21no—
26C2×#386—21no—
27#2285C2212yes—
28C2×#383C2211yescharacteristic
29#2280C2212yes—
30#2280C2212yes—
31C2×#381C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#2274—21no—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37C22×#95C2211yescharacteristic
38#1294C2214yes—
39#1294C2214yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48C2×C4×D4C2×C414yes—
49C2×C4×D4C2×C414yes—
50C2×C4×D4C2×C414yes—
51C2×C4×D4D414yes—
52C2×C4×D4D414yes—
53C2×C4×D4Q814yes—
54C2×#117—22no—
55C2×#117C2×C414yes—
56C2×#117C2×C414yescomplemented
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#435—232no—
68#432—24no—
69#432C2×C418yescomplemented
70C2×C4⋊C8—22no—
71#407—24no—
72#407C2×C418yes—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2×C418yescomplemented
77C2×C22⋊C8C2314yes—
78C2×#97—22no—
79#390—28no—
80#388—28no—
81#388—28no—
82#388—28no—
83#387—28no—
84#386—28no—
85#384C2×C414yescomplemented
86#383—22no—
87#383C2×C414yes—
88#382—22no—
89#381—22no—
90#381—22no—
91#381—22no—
92#381C2×C414yes—
93#380—22no—
94#380—22no—
95#380C2×C414yes—
96#380C2×C414yes—
97#380C2×C414yescomplemented
98#380C2314yes—
99#379—22no—
100#379—22no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C414yescomplemented
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—28no—
110C4×C22⋊C4C2×C414yes—
111#376—22no—
112#376C2×C414yes—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95C2×C414yes—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2311yescharacteristic
123C2×#95C2311yescharacteristic
124C25C2×D411yescharacteristic
125C25C2×D411yescharacteristic
126C22×D4—42no—
127C22×D4—42no—
128C22×D4C22⋊C414yes—
129C22×D4C22⋊C414yes—
130C22×D4C4⋊C414yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×Q812yes—
156C23×C4C2×Q812yes—
157C23×C4C2411yescharacteristic
158C2×M16—28no—
159C2×M16—84no—
160C22×C8—24no—
161C22×C8—28no—
162C22×C8—84no—
163C4×D4C42116yes—
164C4×D4C22⋊C4116yes—
165C4×D4C4⋊C4116yes—
166C4×D4C4⋊C4116yes—
167#117—44no—
168#117—44no—
169#117—84no—
170#117—84no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C22⋊C414yes—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C4⋊C414yes—
178C2×C4⋊C4C22×C418yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4C22⋊C414yes—
191C2×C22⋊C4C22⋊C414yes—
192C2×C22⋊C4C22⋊C414yes—
193C2×C22⋊C4C22⋊C414yes—
194C2×C22⋊C4C4⋊C414yes—
195C2×C22⋊C4C4⋊C414yes—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—44no—
202C2×C42—82no—
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×Q812yes—
209C4⋊C8—28no—
210C22⋊C8—28no—
211C22⋊C8—28no—
212C22⋊C8—24no—
213C22⋊C8C22×C418yes—
214#97—28no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—24no—
228#95C22×C412yes—
229#95C22×C412yes—
230#95C22×C412yes—
231#95C22×C412yes—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C24C2×C22⋊C412yes—
244C24C2×C4⋊C412yes—
245C24C2×C4⋊C412yes—
246C24C22×D411yescharacteristic
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—48no—
252C2×D4#95116yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—84no—
291C22×C4—816no—
292C22×C4C2×C4214yes—
293C22×C4C2×C4214yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C414yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C4⋊C412yes—
305C22×C4C2×C4⋊C414yes—
306C22×C4C2×C4⋊C414yes—
307C22×C4C2×C4⋊C414yes—
308C22×C4C2×C4⋊C414yes—
309C22×C4C2×C4⋊C412yes—
310C22×C4C2×C4⋊C412yes—
311C22×C4C2×C4⋊C412yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C23×C411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318M16—88no—
319C2×C8—28no—
320C2×C8—48no—
321C2×C8—88no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4#95116yes—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C22⋊C4#95116yes—
337C22⋊C4#95116yes—
338C42—48no—
339C42—48no—
340C42—84no—
341C42—84no—
342C42—84no—
343C42C2×C4214yes—
344C42C2×C22⋊C414yes—
345C42C2×C4⋊C414yes—
346C42C2×C4⋊C414yes—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—48no—
370C23—48no—
371C23—84no—
372C23—84no—
373C23C2×#9514yes—
374C23C2×#9514yes—
375C23C22×C22⋊C411yescharacteristic
376C23C22×C22⋊C411yescharacteristic
377C23C22×C4⋊C411yescharacteristic
378C23#57512yes—
379C23#57512yes—
380D4—416no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—816no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4C2×#9514yes—
430C2×C4C2×#9514yes—
431C2×C4C2×#9514yes—
432C2×C4C2×#9514yes—
433C2×C4C22×C4211yescharacteristic
434C2×C4C22×C22⋊C411yescharacteristic
435C2×C4C22×C4⋊C411yescharacteristic
436C2×C4C22×C4⋊C411yescharacteristic
437C8—48no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—48no—
453C22—48no—
454C22—48no—
455C22—82no—
456C22—82no—
457C22—84no—
458C22C22×#9511yescharacteristic
459C22#288312yes—
460C22#289611yescharacteristic
461C22#289611yescharacteristic
462C22#289612yes—
463C22#290112yes—
464C22#291112yes—
465C4—24no—
466C4—24no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—24no—
472C4—24no—
473C4—44no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—88no—
478C2—21no—
479C2—21no—
480C2—22no—
481C2—44no—
482C2—44no—
483C2#1762914yes—
484C2#2034311yescharacteristic
485C2#2034711yescharacteristic
486C2#2035411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8024C2semidirect
5. #8022C2semidirect
6. #8020C2semidirect
7. #8018C2semidirect
8. #8016C2semidirect
9. #8012C2semidirect
12. #2973C4semidirect
18. #2354C4semidirect
20. #2353C4semidirect
56. C2×#117C2×C4semidirect
69. #432C2×C4semidirect
76. C2×C22⋊C8C2×C4semidirect
85. #384C2×C4semidirect
97. #380C2×C4semidirect
103. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#341862
C2×#23532
#80242
#80222
#80202
#80182
#80162
#80122