Finite Group Databasebeta

← #239504

Subgroups — #239504

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5C2×#1461C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic
7#10401C218yesmaximal
8#8026C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2855—22no—
14#2627C2214yes—
15#2582—22no—
16#2581—22no—
17#2568—21no—
18#2567—21no—
19#2565—22no—
20#2317—21no—
21#2310—21no—
22C2×#391—21no—
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1987—21no—
27#1981—21no—
28#1948—22no—
29#1872C2214yes—
30#1853C2214yes—
31C4×#106—22no—
32C4×#103—22no—
33#1518—24no—
34C2×#328C2214yes—
35#1461—24no—
36C2×#324C2214yes—
37#1441C2214yes—
38C4⋊#95—21no—
39#1434—22no—
40#1294C2214yes—
41C23×Q8D411yescharacteristic
42#533—22no—
43#533—22no—
44#533—22no—
45#533—22no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8D412yes—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42—21no—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431—22no—
73#431D414yes—
74C2×#106—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×#103—24no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#400—24no—
83#398—28no—
84#395—24no—
85#391—22no—
86#391—22no—
87#391—24no—
88#391—24no—
89#391—44no—
90#389—24no—
91#389—24no—
92#387—28no—
93#385—24no—
94#384—22no—
95#384D412yes—
96#382—22no—
97#382—21no—
98#382—22no—
99#382—22no—
100#381—24no—
101#380—21no—
102#380C2312yes—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95C2312yes—
114#340—24no—
115#340—24no—
116#328—28no—
117#324—28no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8SD1614yes—
127C22×Q8C2×D412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141#125—42no—
142#125—42no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—42no—
162#117—42no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#106—48no—
212C4⋊C8—24no—
213C4⋊C8—42no—
214#103—48no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8C2×D414yes—
219C4×C8—24no—
220C4×C8—42no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—48no—
229#95—41no—
230#95—41no—
231#95—42no—
232#95—48no—
233C24C22×D411yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4C2×SD1614yes—
285C22×C4C2×SD1612yes—
286C22×C4#13512yes—
287C22×C4#13712yes—
288C22×C4#13712yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—24no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—42no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—88no—
321C4⋊C4—84no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C42—44no—
327C42—44no—
328C42—44no—
329C42—44no—
330C42—82no—
331C42—82no—
332C42#12112yes—
333C42#12112yes—
334C42#12112yes—
335C42#12112yes—
336C42#12312yes—
337C42#12312yes—
338C42#12412yes—
339C42#12412yes—
340C23—22no—
341C23—22no—
342C23C2×#12011yescharacteristic
343C23C2×#12111yescharacteristic
344C23C2×#12311yescharacteristic
345C23C22×SD1611yescharacteristic
346C23C2×#13511yescharacteristic
347C23C2×#13711yescharacteristic
348C23C2×#13711yescharacteristic
349Q8—44no—
350Q8—48no—
351Q8—48no—
352Q8—82no—
353Q8—84no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#39412yes—
395C2×C4#39412yes—
396C2×C4#39412yes—
397C2×C4#39412yes—
398C2×C4#44812yes—
399C2×C4#44812yes—
400C2×C4#45212yes—
401C2×C4#45212yes—
402C2×C4#46514yes—
403C2×C4#47014yes—
404C2×C4#48314yes—
405C2×C4#48514yes—
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×#12311yescharacteristic
409C2×C4C2×#12411yescharacteristic
410C8—44no—
411C8—82no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22C2×#39411yescharacteristic
421C22C2×#44811yescharacteristic
422C22C2×#45211yescharacteristic
423C22#302012yes—
424C22#302512yes—
425C22C2×#46511yescharacteristic
426C22C2×#47011yescharacteristic
427C22C2×#48311yescharacteristic
428C22C2×#48511yescharacteristic
429C4—22no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—84no—
437C4—84no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—162no—
442C4#203514yes—
443C4#203514yes—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2#1821512yes—
449C2#1822012yes—
450C2C2×#203511yescharacteristic
451C2#2349411yescharacteristic
452C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351792
#213852
#213592
C2×#15182
C2×#14612
#139422
#104012
#80262