Finite Group Databasebeta

← #117147

Subgroups — #117147

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21804C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—22no—
11C2×#423C2211yescharacteristic
12C22×#103—22no—
13C2×C4×M16C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2285C2211yescharacteristic
16C2×#384C2211yescharacteristic
17#1997—22no—
18#1997C414yescomplemented
19#1981—22no—
20#1956—22no—
21#1956—21no—
22#1956—21no—
23#1956—22no—
24#1847—22no—
25#1845—21no—
26#1844—21no—
27#1787—22no—
28C2×#343—22no—
29#1296—22no—
30#1279—22no—
31#1278—22no—
32C23×Q8D412yes—
33C22×M16—21no—
34C22×M16D412yes—
35C23×C8—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⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#428—44no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×#103—28no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—48no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16C2×C412yes—
78C4×M16D412yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8D412yes—
88#384—21no—
89#384—21no—
90#384—21no—
91#384D411yescharacteristic
92#384D411yescharacteristic
93#384D412yes—
94#382—21no—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99#343—28no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—44no—
106C22×Q8—44no—
107C22×Q8C4⋊C414yes—
108C23×C4—22no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16C22⋊C414yes—
121C2×M16SD1614yes—
122C2×M16Q1614yes—
123C22×C8—22no—
124C22×C8—24no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—42no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147#117—42no—
148#117—42no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191#107—48no—
192#106—48no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199#103—416no—
200#103—84no—
201#103—84no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—22no—
205#97—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214#95—41no—
215#95—41no—
216C24#12111yescharacteristic
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—48no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—84no—
228C2×Q8—84no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12711yescharacteristic
262C22×C4C2×SD1612yes—
263C22×C4C2×Q1612yes—
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266M16—24no—
267M16—24no—
268M16—24no—
269M16—44no—
270M16—82no—
271M16#10318yescomplemented
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—81no—
285C2×C8—81no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—82no—
308C42—81no—
309C42—81no—
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42#12311yescharacteristic
323C42#12311yescharacteristic
324C42#12411yescharacteristic
325C42#12411yescharacteristic
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#38011yescharacteristic
331C23#39211yescharacteristic
332C23#39711yescharacteristic
333C23#46012yes—
334C23#46212yes—
335Q8—48no—
336Q8—48no—
337Q8—84no—
338Q8—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—164no—
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#39011yescharacteristic
372C2×C4#39311yescharacteristic
373C2×C4#39411yescharacteristic
374C2×C4#39411yescharacteristic
375C2×C4#39611yescharacteristic
376C2×C4#39611yescharacteristic
377C2×C4#39911yescharacteristic
378C2×C4C2×#10312yes—
379C2×C4#41712yes—
380C2×C4#45012yes—
381C2×C4#45112yes—
382C2×C4#45312yes—
383C2×C4#45512yes—
384C2×C4#47112yes—
385C2×C4#47312yes—
386C2×C4#47512yes—
387C2×C4#47712yes—
388C2×C4#48312yes—
389C2×C4#48412yes—
390C2×C4#48712yes—
391C2×C4#48912yes—
392C2×C4#49211yescharacteristic
393C2×C4#49411yescharacteristic
394C2×C4#49611yescharacteristic
395C2×C4#49711yescharacteristic
396C8—28no—
397C8—44no—
398C8—82no—
399C8—82no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#143911yescharacteristic
409C22#180312yes—
410C22#200211yescharacteristic
411C22#200411yescharacteristic
412C22#201411yescharacteristic
413C22#201611yescharacteristic
414C22#201911yescharacteristic
415C22#202011yescharacteristic
416C22#207212yes—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—84no—
422C4—161no—
423C4—162no—
424C4—161no—
425C4#185612yes—
426C4#187212yes—
427C4#193012yes—
428C4#195611yescharacteristic
429C4#195911yescharacteristic
430C4#202412yes—
431C4#203712yes—
432C4#206412yes—
433C4#209511yescharacteristic
434C4#209611yescharacteristic
435C2—22no—
436C2—22no—
437C2#1014611yescharacteristic
438C2#1015511yescharacteristic
439C2#1016311yescharacteristic
440C2#1017011yescharacteristic
441C2#1105011yescharacteristic
442C2#1113511yescharacteristic
443C2#1114511yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
18. #1997C4semidirect
271. M16#103semidirect

Maximal subgroups

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

subgroupindex
#218042
C2×#19972
C2×#19562
#208002
#80432
#78972
C2×#12782