Finite Group Databasebeta

← #393149

Subgroups — #393149

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1951C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic
3#21034C212yesmaximal
4#20331C211yesmaximal, characteristic
5#10913C212yesmaximal
6#10896C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10870C211yesmaximal, characteristic
9#10869C211yesmaximal, characteristic
10#10865C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C2×#389C2212yes—
24C4×#117C2211yescharacteristic
25#1951C418yescomplemented
26#1944—21no—
27#1944—21no—
28#1944—21no—
29#1944—21no—
30#1921—22no—
31#1920—22no—
32#1919—22no—
33#1918—22no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42C2×#340C2212yes—
43#1728C2212yes—
44#1726—21no—
45#1726—21no—
46C4×#97—21no—
47C4×#97—21no—
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60C2×#108—22no—
61C2×#108—22no—
62C2×#108—22no—
63C2×#108—22no—
64C2×#108—22no—
65C2×#108C2×C414yes—
66#428—24no—
67#428C2×C418yes—
68C2×#107—22no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106C2×C414yes—
78#405—22no—
79#405—22no—
80#405—22no—
81#405—22no—
82#405—22no—
83#405C2×C414yes—
84#405Q814yes—
85#405Q814yes—
86C4×M16—22no—
87C4×M16—22no—
88C2×#97—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—21no—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97—21no—
95C2×#97D411yescharacteristic
96C2×#97D411yescharacteristic
97C2×#97D411yescharacteristic
98C2×#97D411yescharacteristic
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—21no—
113C2×#95—22no—
114C43—21no—
115#343—24no—
116#343—24no—
117#343—24no—
118#343C2×C418yes—
119#341—24no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#336—22no—
129#336—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
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×M16C22×C414yes—
153C2×M16C22×C412yes—
154C2×M16C2×Q812yes—
155C2×M16C2×Q812yes—
156C2×M16#4112yes—
157C2×M16#4112yes—
158C2×M16#4112yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q812yes—
173C22×C8C2×Q812yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178C22×C8#4112yes—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117C22×C412yes—
184#117C2×Q812yes—
185#117C2×Q812yes—
186#117#4112yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218#108—28no—
219#108—44no—
220#107—28no—
221#107—42no—
222#107—42no—
223#106—24no—
224#106—24no—
225#106—28no—
226#97—24no—
227#97—24no—
228#97—24no—
229#97—24no—
230#97—24no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97—22no—
235C4×C8—24no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—24no—
240#95—22no—
241#95—22no—
242#95—42no—
243#95—42no—
244C24C2×#4111yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C22×Q811yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281M16—22no—
282M16—22no—
283M16—22no—
284M16—22no—
285M16—22no—
286M16—22no—
287M16—24no—
288M16—24no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16C4×Q814yes—
296M16#12514yes—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8#11714yes—
318C2×C8C4×D412yes—
319C2×C8C4×D412yes—
320C2×C8C4×Q814yes—
321C2×C8C22⋊Q812yes—
322C2×C8C22⋊Q812yes—
323C2×C8C22⋊Q812yes—
324C2×C8C22⋊Q812yes—
325C2×C8#12312yes—
326C2×C8#12312yes—
327C2×C8#12514yes—
328C2×C8#12614yes—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4C4×D414yes—
342C4⋊C4C4⋊Q814yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—22no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—82no—
360C42—82no—
361C23—22no—
362C23—22no—
363C23C4×#4111yescharacteristic
364C23C2×#12111yescharacteristic
365C23#52911yescharacteristic
366C23#52911yescharacteristic
367C23#52911yescharacteristic
368C23#53311yescharacteristic
369C23#53311yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—162no—
397C2×C4—162no—
398C2×C4C2×#11711yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×C4×Q811yescharacteristic
402C2×C4C2×C4×Q811yescharacteristic
403C2×C4C4×#4111yescharacteristic
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×#12311yescharacteristic
407C2×C4#52511yescharacteristic
408C2×C4#52511yescharacteristic
409C2×C4C2×#12511yescharacteristic
410C2×C4C2×#12511yescharacteristic
411C2×C4C2×#12611yescharacteristic
412C2×C4#52911yescharacteristic
413C2×C4#52911yescharacteristic
414C2×C4#52911yescharacteristic
415C2×C4#52911yescharacteristic
416C2×C4#52911yescharacteristic
417C2×C4C2×C4⋊Q811yescharacteristic
418C2×C4#53211yescharacteristic
419C2×C4#53211yescharacteristic
420C2×C4#53211yescharacteristic
421C2×C4#53211yescharacteristic
422C2×C4#53311yescharacteristic
423C2×C4#53311yescharacteristic
424C2×C4#53311yescharacteristic
425C2×C4#53311yescharacteristic
426C8—24no—
427C8—22no—
428C8—22no—
429C8—24no—
430C8—44no—
431C8—82no—
432C8#38714yes—
433C8#38714yes—
434C22—22no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22#229511yescharacteristic
443C22C4×#12111yescharacteristic
444C22C4×#12511yescharacteristic
445C22#285411yescharacteristic
446C22#285711yescharacteristic
447C22#286311yescharacteristic
448C22#286411yescharacteristic
449C22#306812yes—
450C22#306912yes—
451C4—44no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—44no—
457C4—84no—
458C4—162no—
459C4C2×#38711yescharacteristic
460C4#229311yescharacteristic
461C4C4×C22⋊Q811yescharacteristic
462C4C4×C4⋊Q811yescharacteristic
463C4#285111yescharacteristic
464C4#285511yescharacteristic
465C4#285911yescharacteristic
466C4#286011yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2#1469911yescharacteristic
472C2#2155712yes—
473C2#3312511yescharacteristic
474C2#3312611yescharacteristic
475C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1951C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19512
C2×#19442
#210342
#203312
#109132
#108962
#108702
#108692
#108652
#108272