Finite Group Databasebeta

← #198685

Subgroups — #198685

Order 512.

1043 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#21799C211yesmaximal, characteristic
4#17749C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9272C218yesmaximal, complemented
7#8076C211yesmaximal, characteristic
8#7934C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3229—22no—
12#3229—22no—
13#3229C414yes—
14#3229C2214yes—
15C2×#498—21no—
16C2×#487—21no—
17C2×#483C414yes—
18C2×#475—22no—
19#2986—22no—
20#2985—22no—
21#2984—22no—
22#2983—21no—
23#2982—21no—
24#2962C2214yes—
25#2960—22no—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423C2211yescharacteristic
32#1966—22no—
33#1966C2214yes—
34#1784—22no—
35#1559—22no—
36#1550C2214yes—
37#1549—21no—
38#1549—21no—
39#1549—22no—
40#1548—21no—
41#1546—21no—
42C2×#332C2214yes—
43#1301C2214yes—
44#1283C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533C2×C412yes—
52#533D412yes—
53#533D412yes—
54#533C2312yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2312yes—
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2×C412yes—
72C2×C4×Q8—21no—
73C2×C4×Q8D412yes—
74C2×#117—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C412yes—
77C22×C42C2311yescharacteristic
78#498—24no—
79#498—24no—
80#487—24no—
81#487—24no—
82#483—28no—
83#483C2×C4116yes—
84#475—28no—
85#475—28no—
86#446—24no—
87#446—24no—
88#446—28no—
89#446—24no—
90C8×Q8—24no—
91C8×Q8—24no—
92C8×Q8—24no—
93#433—28no—
94#432—24no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#431—24no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431D411yescharacteristic
110#431D411yescharacteristic
111#431D412yes—
112#431C2312yes—
113C2×#106—24no—
114#424—24no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423D412yes—
120#423C2312yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103—24no—
126C2×#101—28no—
127#407—21no—
128#407D412yes—
129C2×C22⋊C8—21no—
130C2×C22⋊C8C2×C412yes—
131#405—24no—
132C4×M16—21no—
133C4×M16C2312yes—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136#333—24no—
137#332—24no—
138#332—24no—
139#332—24no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—84no—
148C22×Q8C22⋊C412yes—
149C23×C4—22no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—22no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16—82no—
157C2×M16C2×D414yes—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8C22⋊C412yes—
182C4⋊Q8C22⋊C412yes—
183C4⋊Q8C2×D412yes—
184C4⋊Q8C2×D412yes—
185#125—22no—
186#125—22no—
187#125—22no—
188#125—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8C22⋊C414yes—
196C22⋊Q8C22×C414yes—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C22⋊C414yes—
204C4×Q8C2×D414yes—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22×C414yes—
227C2×C22⋊C4—22no—
228C2×C42—22no—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241#106—216no—
242C4⋊C8—28no—
243C4⋊C8—28no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8C2×D412yes—
256C4⋊C8C2×D412yes—
257#103—216no—
258#101—88no—
259C22⋊C8—22no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8C22×C414yes—
268#97—28no—
269#97—22no—
270#97—41no—
271C4×C8—28no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—22no—
275C4×C8—21no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—41no—
279C24C4×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—44no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—48no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8C2×C22⋊C414yes—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—84no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325M16—28no—
326M16—44no—
327M16—82no—
328M16—82no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—82no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4C2×C22⋊C414yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C42—24no—
361C42—24no—
362C42C2×C22⋊C411yescharacteristic
363C42C2×C22⋊C411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42#12011yescharacteristic
367C42#12011yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12311yescharacteristic
373C42#12311yescharacteristic
374C42C22×D411yescharacteristic
375C42C22×D411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C23—22no—
379C23—22no—
380C23—41no—
381C23#38611yescharacteristic
382C23#38611yescharacteristic
383C23C2×C4×D411yescharacteristic
384Q8—44no—
385Q8—44no—
386Q8—44no—
387Q8—42no—
388Q8—42no—
389Q8—84no—
390Q8—84no—
391Q8—84no—
392Q8—84no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4#38611yescharacteristic
420C2×C4#38611yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#47112yes—
426C2×C4#47312yes—
427C2×C4C22×C22⋊C411yescharacteristic
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4#57714yes—
434C8—44no—
435C8—42no—
436C8—42no—
437C8—44no—
438C8—82no—
439C8—82no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—44no—
448C22#190812yes—
449C22C2×#38611yescharacteristic
450C22#294912yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—44no—
459C4—44no—
460C4—84no—
461C4#190812yes—
462C4#191512yes—
463C4#301012yes—
464C4#304312yes—
465C4C2×#47111yescharacteristic
466C4C2×#47311yescharacteristic
467C2—22no—
468C2—22no—
469C2—41no—
470C2#2093611yescharacteristic
471C2C2×#190811yescharacteristic
472C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9272C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#602502
#217992
#177492
C2×#15492
#92722
#80762
#79342