Finite Group Databasebeta

← #193307

Subgroups — #193307

Order 512.

912 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
1#34374C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21961C211yesmaximal, characteristic
4#21835C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8923C212yesmaximal, complemented
8#8922C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7900C211yesmaximal, characteristic
11#7893C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2382—22no—
17#2365C2212yes—
18#2365C2212yes—
19#2355—21no—
20#2355—21no—
21Q8×C4⋊C4—21no—
22Q8×C4⋊C4—21no—
23#2323—22no—
24#2323—22no—
25#2314—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1989—22no—
34#1948C2212yes—
35#1947—21no—
36#1947—21no—
37#1947C2212yes—
38#1947C2212yes—
39#1947C2212yes—
40#1854—22no—
41#1847—21no—
42#1836—21no—
43#1825—22no—
44#1808—22no—
45#1552C418yescomplemented
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×C8—21no—
53#533—22no—
54#533—22no—
55#533C2×C414yes—
56#533Q814yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125D412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D412yes—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432C2×C414yes—
73#431—22no—
74#431—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C414yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×#97—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#391—22no—
92#389—24no—
93#389—24no—
94#389—22no—
95#389—22no—
96#388—24no—
97#388—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#384—22no—
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384D411yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—22no—
110#382—21no—
111#382—21no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380D412yes—
118#380D412yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#332C2×C418yes—
138#332C2×C418yes—
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—24no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—81no—
150C22×C8—81no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8C22×C412yes—
156C4⋊Q8C22×C412yes—
157C4⋊Q8C2×Q812yes—
158C4⋊Q8C2×Q812yes—
159#125—24no—
160#125C4⋊C418yes—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—82no—
170C4×Q8—82no—
171C4×Q8—82no—
172C4×Q8—82no—
173#117—24no—
174#117—82no—
175#117—81no—
176#117—81no—
177#117—82no—
178#117C4⋊C418yes—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—41no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—82no—
224#107—82no—
225#107—82no—
226#106—82no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C22×C414yes—
232C4⋊C8C22×C414yes—
233C22⋊C8—24no—
234C22⋊C8—42no—
235#97—24no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—41no—
249#95—41no—
250C24#12011yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—41no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4C2×C4⋊C412yes—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4C22⋊Q812yes—
284C22×C4C22⋊Q812yes—
285C22×C4#12312yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—84no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—81no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4C2×C4⋊C414yes—
336C4⋊C4C2×C4⋊C414yes—
337C22⋊C4—28no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—28no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—82no—
347C42—84no—
348C42—84no—
349C42C2×C4⋊C412yes—
350C42C4×D412yes—
351C42C4×D412yes—
352C42C22⋊Q812yes—
353C42C22⋊Q812yes—
354C42#12312yes—
355C42C23×C411yescharacteristic
356C42C22×Q811yescharacteristic
357C42C2×#4111yescharacteristic
358C42C2×#4111yescharacteristic
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#38512yes—
366C23C2×#12011yescharacteristic
367Q8—42no—
368Q8—44no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—44no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#38512yes—
414C2×C4#38512yes—
415C2×C4#38512yes—
416C2×C4C22×C4⋊C411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×C4×D411yescharacteristic
419C2×C4C2×C22⋊Q811yescharacteristic
420C2×C4C2×C22⋊Q811yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4#57611yescharacteristic
423C2×C4#57612yes—
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C2×C4#57812yes—
428C8—44no—
429C8—44no—
430C8—81no—
431C8—81no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22C2×#38511yescharacteristic
440C22#300811yescharacteristic
441C22#301111yescharacteristic
442C22#301211yescharacteristic
443C22#301311yescharacteristic
444C22#302011yescharacteristic
445C22#302011yescharacteristic
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—161no—
458C4—161no—
459C4#294611yescharacteristic
460C4#294711yescharacteristic
461C4#295112yes—
462C4#308311yescharacteristic
463C4#308312yes—
464C4#308411yescharacteristic
465C2—42no—
466C2#1894611yescharacteristic
467C2#1894811yescharacteristic
468C2#1895111yescharacteristic
469C2#1895311yescharacteristic
470C2#2085711yescharacteristic
471C2#2087811yescharacteristic
472C2#2088111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8923C2semidirect
8. #8922C2semidirect
9. #8922C2semidirect
45. #1552C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219882
#219612
#218352
C2×#15522
#89262
#89232
#89222
#79002
#78932