Finite Group Databasebeta

← #117150

Subgroups — #117150

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21833C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1966C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×#101—22no—
12C2×C4×M16C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#1990—22no—
17#1990—22no—
18#1990—21no—
19#1990—21no—
20#1988—22no—
21#1966—22no—
22#1966C414yescomplemented
23#1836—22no—
24#1834—21no—
25#1834—21no—
26#1823—22no—
27#1736—22no—
28C2×#340—22no—
29#1296—22no—
30#1279—22no—
31#1278—22no—
32C23×Q8—21no—
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⋊C4D412yes—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423C2×C412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D412yes—
66C2×#101—28no—
67C2×#101—44no—
68C2×#101—44no—
69C2×C22⋊C8—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C4×M16D412yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8D412yes—
83#384—21no—
84#384—21no—
85#384—21no—
86#384D411yescharacteristic
87#384D411yescharacteristic
88#384D412yes—
89#382—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—28no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C23×C4—22no—
110C23×C4—41no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—41no—
121C2×M16—42no—
122C2×M16—42no—
123C2×M16C22⋊C414yes—
124C22×C8—24no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—42no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
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×C4⋊C4C4⋊C414yes—
178C2×C22⋊C4—41no—
179C2×C42—41no—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×Q811yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198#101—416no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#97—22no—
202#97—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—41no—
211#95—41no—
212#95—41no—
213C24#12111yescharacteristic
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—81no—
229C2×Q8—81no—
230C2×Q8—81no—
231C2×Q8—81no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4#11711yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12711yescharacteristic
265C22×C4#13512yes—
266C22×C4#13512yes—
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4AGL(1,ℤ/8)12yes—
269M16—24no—
270M16—24no—
271M16—42no—
272M16—42no—
273M16—44no—
274M16—82no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—81no—
288C2×C8—81no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C22⋊C4—82no—
311C42—81no—
312C42—81no—
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42#12111yescharacteristic
318C42#12111yescharacteristic
319C42#12111yescharacteristic
320C42#12111yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12411yescharacteristic
328C42#12411yescharacteristic
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#38011yescharacteristic
334C23#39211yescharacteristic
335C23#39711yescharacteristic
336C23#46312yes—
337C23#46312yes—
338Q8—82no—
339Q8—82no—
340Q8—84no—
341Q8—82no—
342Q8—82no—
343Q8—84no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—164no—
371C2×C4#38511yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38611yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#38811yescharacteristic
376C2×C4#39011yescharacteristic
377C2×C4#39311yescharacteristic
378C2×C4#39411yescharacteristic
379C2×C4#39411yescharacteristic
380C2×C4#39611yescharacteristic
381C2×C4#39611yescharacteristic
382C2×C4#39911yescharacteristic
383C2×C4#41612yes—
384C2×C4#41812yes—
385C2×C4#44912yes—
386C2×C4#44912yes—
387C2×C4#45412yes—
388C2×C4#45612yes—
389C2×C4#47112yes—
390C2×C4#47312yes—
391C2×C4#47812yes—
392C2×C4#47812yes—
393C2×C4#48212yes—
394C2×C4#48212yes—
395C2×C4#48812yes—
396C2×C4#49012yes—
397C2×C4#49511yescharacteristic
398C2×C4#49511yescharacteristic
399C2×C4#49611yescharacteristic
400C2×C4#49611yescharacteristic
401C8—44no—
402C8—44no—
403C8—82no—
404C8—82no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#143911yescharacteristic
414C22#180412yes—
415C22#200311yescharacteristic
416C22#200511yescharacteristic
417C22#201211yescharacteristic
418C22#201211yescharacteristic
419C22#201911yescharacteristic
420C22#202011yescharacteristic
421C22#207312yes—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—84no—
427C4—162no—
428C4—161no—
429C4—161no—
430C4#185712yes—
431C4#187512yes—
432C4#193012yes—
433C4#195811yescharacteristic
434C4#196111yescharacteristic
435C4#202512yes—
436C4#203812yes—
437C4#206412yes—
438C4#209211yescharacteristic
439C4#209211yescharacteristic
440C2—22no—
441C2—22no—
442C2#1015111yescharacteristic
443C2#1015811yescharacteristic
444C2#1016611yescharacteristic
445C2#1017511yescharacteristic
446C2#1105111yescharacteristic
447C2#1113611yescharacteristic
448C2#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
22. #1966C4semidirect

Maximal subgroups

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

subgroupindex
#218332
C2×#19902
C2×#19662
#207642
#80432
#78972
C2×#12782