Finite Group Databasebeta

← #118646

Subgroups — #118646

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21321C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#10843C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7C2×#1281C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#97C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993—21no—
16#1993—21no—
17#1992—21no—
18#1992—21no—
19#1990—21no—
20#1956—21no—
21#1947—22no—
22#1872—22no—
23#1852—22no—
24#1845—21no—
25#1834—21no—
26#1803—22no—
27#1735—22no—
28C2×#340—22no—
29#1518—22no—
30#1518C414yescomplemented
31C2×#328—22no—
32#1294—22no—
33#1281—22no—
34C23×C8D412yes—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65C2×#107—24no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×#103—24no—
69C2×C22⋊C8D412yes—
70C2×C22⋊C8D412yes—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384D412yes—
83#384D411yescharacteristic
84#384D411yescharacteristic
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382C2×C412yes—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C2×#95—21no—
95C2×#95—21no—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—44no—
103#328—28no—
104C22×Q8—21no—
105C22×Q8—21no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8Q1614yes—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—41no—
151#117—41no—
152#117—42no—
153#117—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196#107—84no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201#103—84no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8C22⋊C414yes—
206#97—41no—
207#97—41no—
208#97—41no—
209#97—41no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214#95—22no—
215#95—22no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24#12711yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—81no—
228C2×Q8—81no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12411yescharacteristic
267C22×C4C2×Q1612yes—
268C22×C4#13512yes—
269C22×C4AGL(1,ℤ/8)12yes—
270C22×C4#13712yes—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—42no—
315C42—42no—
316C42—82no—
317C42C2×C4⋊C411yescharacteristic
318C42#11711yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12411yescharacteristic
330C42#12411yescharacteristic
331C42#12611yescharacteristic
332C42#12611yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#39011yescharacteristic
338C23#39211yescharacteristic
339C23#39911yescharacteristic
340C23#49411yescharacteristic
341C23#49511yescharacteristic
342C23#49611yescharacteristic
343C23#49711yescharacteristic
344Q8—44no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—161no—
383C2×C4—161no—
384C2×C4—161no—
385C2×C4—161no—
386C2×C4—162no—
387C2×C4—162no—
388C2×C4#38011yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38511yescharacteristic
391C2×C4#38611yescharacteristic
392C2×C4#38811yescharacteristic
393C2×C4#38811yescharacteristic
394C2×C4#39311yescharacteristic
395C2×C4#39411yescharacteristic
396C2×C4#39411yescharacteristic
397C2×C4#39611yescharacteristic
398C2×C4#39611yescharacteristic
399C2×C4#39711yescharacteristic
400C2×C4C4×Q1612yes—
401C2×C4#44012yes—
402C2×C4#44912yes—
403C2×C4#45112yes—
404C2×C4#46212yes—
405C2×C4#46312yes—
406C2×C4#46712yes—
407C2×C4#46812yes—
408C2×C4#47712yes—
409C2×C4#47912yes—
410C2×C4#48412yes—
411C2×C4#48512yes—
412C8—44no—
413C8—44no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22#143911yescharacteristic
428C22#196911yescharacteristic
429C22#197311yescharacteristic
430C22#200811yescharacteristic
431C22#200811yescharacteristic
432C22#200911yescharacteristic
433C22#200911yescharacteristic
434C22#201211yescharacteristic
435C22#201611yescharacteristic
436C22#209511yescharacteristic
437C22#209711yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—82no—
447C4—82no—
448C4—162no—
449C4—161no—
450C4—162no—
451C4—161no—
452C4#186312yes—
453C4#192812yes—
454C4#203612yes—
455C4#206812yes—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#1021911yescharacteristic
461C2#1022411yescharacteristic
462C2#1022411yescharacteristic
463C2#1022611yescharacteristic
464C2#1109311yescharacteristic
465C2#1111711yescharacteristic
466C2#1112411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
30. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#219822
#218602
#213212
C2×#15182
#108432
#80262
C2×#12812