Finite Group Databasebeta

← #117454

Subgroups — #117454

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#10839C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#391—22no—
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C43C2211yescharacteristic
16#1990—21no—
17#1990—21no—
18#1987—22no—
19#1948—22no—
20#1853—22no—
21#1844—21no—
22#1834—21no—
23#1824—22no—
24C2×#339—22no—
25#1549—22no—
26#1549C414yescomplemented
27C2×#332—22no—
28C4⋊#95—22no—
29C4⋊#95—22no—
30#1296—22no—
31#1276—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8Q812yes—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64C2×#106—24no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D412yes—
72C2×C4⋊C8D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—48no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384D411yescharacteristic
83#384D411yescharacteristic
84#382—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C2×#95—28no—
90C43—21no—
91C43—21no—
92C43—21no—
93C43—21no—
94#340—24no—
95#340—24no—
96#340—24no—
97#339—44no—
98#332—28no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—44no—
102C22×Q8—44no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—41no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—42no—
116C2×M16—42no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136#117—42no—
137#117—42no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C4⋊C414yes—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42SD1614yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194#106—84no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8C22⋊C414yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—41no—
210C4×C8—41no—
211#95—416no—
212#95—41no—
213#95—41no—
214C24#12011yescharacteristic
215C2×Q8—48no—
216C2×Q8—48no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12711yescharacteristic
267C22×C4C2×SD1612yes—
268C22×C4#13512yes—
269C22×C4#13712yes—
270C22×C4#13712yes—
271M16—44no—
272M16—82no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—81no—
281C2×C8—81no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C22⋊C4—82no—
304C42—24no—
305C42—24no—
306C42—44no—
307C42—44no—
308C42—44no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—81no—
313C42—81no—
314C42C2×C22⋊C411yescharacteristic
315C42#11711yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42#12111yescharacteristic
319C42#12111yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12411yescharacteristic
327C42#12411yescharacteristic
328C42#12611yescharacteristic
329C42#12611yescharacteristic
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#38511yescharacteristic
335C23#39211yescharacteristic
336C23#39411yescharacteristic
337C23#44812yes—
338C23#45212yes—
339Q8—84no—
340Q8—84no—
341Q8—84no—
342Q8—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—88no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4#38011yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38611yescharacteristic
380C2×C4#38811yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#39011yescharacteristic
383C2×C4#39311yescharacteristic
384C2×C4#39411yescharacteristic
385C2×C4#39611yescharacteristic
386C2×C4#39611yescharacteristic
387C2×C4#39711yescharacteristic
388C2×C4#39911yescharacteristic
389C2×C4C4×SD1612yes—
390C2×C4#44112yes—
391C2×C4#45312yes—
392C2×C4#45412yes—
393C2×C4#45412yes—
394C2×C4#45612yes—
395C2×C4#46112yes—
396C2×C4#46412yes—
397C2×C4#46512yes—
398C2×C4#47012yes—
399C2×C4#47512yes—
400C2×C4#47912yes—
401C2×C4#48312yes—
402C2×C4#48512yes—
403C2×C4#49211yescharacteristic
404C2×C4#49511yescharacteristic
405C2×C4#49711yescharacteristic
406C2×C4#49711yescharacteristic
407C8—44no—
408C8—82no—
409C8—82no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22#143911yescharacteristic
419C22#186512yes—
420C22#200211yescharacteristic
421C22#200311yescharacteristic
422C22#200311yescharacteristic
423C22#200511yescharacteristic
424C22#201311yescharacteristic
425C22#201511yescharacteristic
426C22#203512yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—48no—
432C4—84no—
433C4—161no—
434C4—161no—
435C4—162no—
436C4#185712yes—
437C4#188312yes—
438C4#192712yes—
439C4#196811yescharacteristic
440C4#197111yescharacteristic
441C4#202712yes—
442C4#203812yes—
443C4#206612yes—
444C4#209611yescharacteristic
445C4#209711yescharacteristic
446C2—22no—
447C2—22no—
448C2#1028311yescharacteristic
449C2#1028511yescharacteristic
450C2#1029211yescharacteristic
451C2#1029611yescharacteristic
452C2#1106311yescharacteristic
453C2#1112211yescharacteristic
454C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#218342
C2×#15492
#147142
#108392
#80432
C2×#12762