Finite Group Databasebeta

← #193280

Subgroups — #193280

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic
4#21365C211yesmaximal, characteristic
5C2×#1559C211yesmaximal, characteristic
6#8927C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8926C212yesmaximal, complemented
9#8923C212yesmaximal, complemented
10#7901C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2370—22no—
17#2369C2212yes—
18#2365C2212yes—
19#2355—21no—
20#2355—22no—
21Q8×C4⋊C4—21no—
22#2329—22no—
23#2317—21no—
24C4×C22⋊Q8—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2274—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948—21no—
34#1948C2212yes—
35#1948C2212yes—
36#1947—21no—
37#1947C2212yes—
38#1947C2212yes—
39#1873—22no—
40#1853—21no—
41#1852—21no—
42#1809—22no—
43#1805—22no—
44#1776—22no—
45#1559C418yescomplemented
46C2×#333C2212yes—
47C2×#332C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×M16—21no—
53#533—22no—
54#533—22no—
55#533C2×C414yes—
56#533Q814yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8D412yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432C2×C414yes—
74#431—22no—
75#431—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#106—22no—
80#423—22no—
81#423C2×C414yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#103—22no—
87C2×#103—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—24no—
94#391—22no—
95#389—24no—
96#389—22no—
97#389—22no—
98#389—22no—
99#387—24no—
100#387—24no—
101#387—24no—
102#385—24no—
103#384—22no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—22no—
109#382—21no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380D412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C2×#95—21no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#333C2×C418yescomplemented
138#332C2×C418yescomplemented
139C22×Q8—21no—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—81no—
148C2×M16—81no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8C22×C412yes—
158C4⋊Q8C22×C412yes—
159C4⋊Q8C2×Q812yes—
160C4⋊Q8C2×Q812yes—
161#125—24no—
162#125—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8C4⋊C418yes—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—82no—
171C4×Q8—82no—
172C4×Q8C4⋊C418yes—
173#117—24no—
174#117—24no—
175#117—44no—
176#117—82no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
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—42no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—41no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—41no—
213C2×C42—81no—
214C2×C42—81no—
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—44no—
224#107—82no—
225#106—44no—
226#106—82no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C22×C414yes—
232C4⋊C8C22×C414yes—
233#103—82no—
234#103—82no—
235C22⋊C8—24no—
236C22⋊C8—42no—
237#97—24no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—41no—
251#95—41no—
252C24#12011yescharacteristic
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8C2×C4⋊C414yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4C2×C4⋊C412yes—
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4C22⋊Q812yes—
292C22×C4C22⋊Q812yes—
293C22×C4#12312yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—84no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4C2×C4⋊C414yes—
343C4⋊C4#13714yes—
344C22⋊C4—28no—
345C22⋊C4—44no—
346C22⋊C4—82no—
347C42—28no—
348C42—44no—
349C42—42no—
350C42—42no—
351C42—82no—
352C42—82no—
353C42—84no—
354C42—84no—
355C42C2×C4⋊C412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C22⋊Q812yes—
359C42C22⋊Q812yes—
360C42#12312yes—
361C42C23×C411yescharacteristic
362C42C22×Q811yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#38512yes—
372C23C2×#12011yescharacteristic
373Q8—44no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378Q8—82no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
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—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#38512yes—
422C2×C4#38512yes—
423C2×C4#38512yes—
424C2×C4C22×C4⋊C411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4C2×#13712yes—
431C2×C4#57512yes—
432C2×C4#57611yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57811yescharacteristic
436C8—44no—
437C8—44no—
438C8—81no—
439C8—81no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22C2×#38511yescharacteristic
448C22#300411yescharacteristic
449C22#300511yescharacteristic
450C22#301211yescharacteristic
451C22#301311yescharacteristic
452C22#302111yescharacteristic
453C22#302211yescharacteristic
454C4—21no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—44no—
460C4—44no—
461C4—44no—
462C4—82no—
463C4—82no—
464C4—82no—
465C4—161no—
466C4—161no—
467C4C4×#13712yes—
468C4#294611yescharacteristic
469C4#294711yescharacteristic
470C4#307812yes—
471C4#308311yescharacteristic
472C4#308411yescharacteristic
473C2—42no—
474C2#1883911yescharacteristic
475C2#1884111yescharacteristic
476C2#1886311yescharacteristic
477C2#1886511yescharacteristic
478C2#2085811yescharacteristic
479C2#2087411yescharacteristic
480C2#2087811yescharacteristic
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. #8927C2semidirect
7. #8927C2semidirect
8. #8926C2semidirect
9. #8923C2semidirect
45. #1559C4semidirect
137. #333C2×C4semidirect
138. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219642
#218642
#213652
C2×#15592
#89272
#89262
#89232
#79012
#79002