Finite Group Databasebeta

← #386111

Subgroups — #386111

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#1997C212yesmaximal
3#21023C212yesmaximal
4#11382C212yesmaximal
5#11354C212yesmaximal
6#11204C212yesmaximal
7#10913C212yesmaximal
8#10802C211yesmaximal, characteristic
9#10802C211yesmaximal, characteristic
10C22×C4⋊Q8C412yes—
11C22×C4⋊Q8C2212yes—
12C2×#423C2212yes—
13C2×C4×M16C2212yes—
14#2551—22no—
15#2551—22no—
16#2551—22no—
17#2551—22no—
18C2×#395C414yes—
19C2×#389C412yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1997—28no—
25#1917—22no—
26#1917—22no—
27#1916—22no—
28#1916—22no—
29C2×#343C2212yes—
30C2×#343C2211yescharacteristic
31C2×#343C2212yes—
32C2×#343C2211yescharacteristic
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39#1728C2212yes—
40#1728C2212yes—
41#1728C2212yes—
42#1728C2212yes—
43C23×Q8C2×C414yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—48no—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42C2×C412yes—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#423—24no—
73#423—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C4×M16—24no—
77C4×M16—42no—
78C2×#97—22no—
79C2×C4×C8—22no—
80#395—24no—
81#395—24no—
82#395—24no—
83#395—24no—
84#389—22no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#384—22no—
92#384—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C414yes—
101C2×#95C2×C412yes—
102#343—24no—
103#343—24no—
104#343—24no—
105#343—22no—
106#343—24no—
107#343—22no—
108#343—42no—
109#343—42no—
110#343D414yes—
111#343Q814yescomplemented
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—28no—
123C22×Q8—44no—
124C22×Q8—48no—
125C22×Q8—84no—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C414yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C2411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16C2×D414yes—
157C2×M16C2×Q814yes—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—416no—
171C4⋊Q8—48no—
172C4⋊Q8—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C4⋊C4#4112yes—
206C2×C4⋊C4#4112yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—44no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C4⋊C8—44no—
224C4⋊C8—44no—
225#97—44no—
226C4×C8—44no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24C23×C411yescharacteristic
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—816no—
242C2×Q8—816no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C4×D414yes—
280C22×C4C4×Q814yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×Q811yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288C22×C42−1+411yescharacteristic
289C22×C42−1+411yescharacteristic
290C22×C42−1+412yes—
291M16—24no—
292M16—24no—
293M16—24no—
294M16—24no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—48no—
336C4⋊C4—44no—
337C4⋊C4—84no—
338C4⋊C4#10014yes—
339C4⋊C4#10114yes—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—44no—
353C42—44no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23C22×C22⋊C411yescharacteristic
359C23C2×C4×D411yescharacteristic
360C23C2×C4×Q811yescharacteristic
361C23#51811yescharacteristic
362C23#51911yescharacteristic
363C23#51911yescharacteristic
364C23#51911yescharacteristic
365Q8—816no—
366Q8—816no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4C2×#10012yes—
408C2×C4C2×#10014yes—
409C2×C4C2×#10112yes—
410C2×C4C2×#10114yes—
411C2×C4#53612yes—
412C2×C4#53612yes—
413C2×C4#53612yes—
414C2×C4#54112yes—
415C2×C4#54212yes—
416C2×C4#54412yes—
417C2×C4D4212yes—
418C2×C4#54612yes—
419C2×C4#54711yescharacteristic
420C2×C4#54812yes—
421C2×C4D4×Q811yescharacteristic
422C2×C4D4×Q812yes—
423C2×C4#55611yescharacteristic
424C2×C4#55711yescharacteristic
425C2×C4#56312yes—
426C2×C4#56312yes—
427C8—44no—
428C8—44no—
429C8—82no—
430C8—82no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22#231011yescharacteristic
444C22#232711yescharacteristic
445C22#233111yescharacteristic
446C22D4×C22⋊C411yescharacteristic
447C22Q8×C22⋊C411yescharacteristic
448C22#236511yescharacteristic
449C22#238111yescharacteristic
450C22C22×#10011yescharacteristic
451C22C22×#10111yescharacteristic
452C22#288811yescharacteristic
453C22#288911yescharacteristic
454C4—22no—
455C4—22no—
456C4—24no—
457C4—24no—
458C4—42no—
459C4—42no—
460C4—44no—
461C4—42no—
462C4—44no—
463C4—84no—
464C4#267412yes—
465C4#267712yes—
466C4#271312yes—
467C4#271612yes—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2#1442011yescharacteristic
473C2#2182911yescharacteristic
474C2#2183211yescharacteristic
475C2D4×#10011yescharacteristic
476C2D4×#10111yescharacteristic
477C2Q8×#10011yescharacteristic
478C2Q8×#10111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
111. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#19972
#210232
#113822
#113542
#112042
#109132
#108022