Finite Group Databasebeta

← #112672

Subgroups — #112672

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic
2#21838C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4C2×#1556C211yesmaximal, characteristic
5C2×#1470C211yesmaximal, characteristic
6C2×#1302C211yesmaximal, characteristic
7#8034C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×#125C2211yescharacteristic
10C2×#491—22no—
11C2×#491—22no—
12C2×#491—22no—
13C2×#491—22no—
14C2×#491—22no—
15C2×#491C414yescomplemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×#103—21no—
20C22×#103—21no—
21C22×#97C2211yescharacteristic
22#2285C2211yescharacteristic
23#1992—22no—
24#1990—22no—
25#1989—22no—
26#1987—22no—
27#1824—22no—
28#1824—21no—
29#1823—21no—
30#1556—22no—
31#1556—22no—
32C2×#332—22no—
33#1470—22no—
34#1470—22no—
35C2×#324—22no—
36#1302—22no—
37#1295—22no—
38C23×Q8—21no—
39C23×C8—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⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8D412yes—
51C2×C4⋊Q8D412yes—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125C2×C412yes—
63C2×#125D412yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72#491—28no—
73#491—28no—
74#491—28no—
75#491—28no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#432—21no—
82#432—21no—
83#432D412yes—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D412yes—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D412yes—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97C2×C412yes—
106C2×#97D412yes—
107C2×#97D412yes—
108C2×C4×C8—21no—
109#384D412yes—
110#382—21no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115#340—24no—
116#340—28no—
117#340—24no—
118#332—28no—
119#324—28no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—48no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—44no—
139C22×C8—41no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8C22⋊C414yes—
156#125—22no—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#125—22no—
162#125—22no—
163#125—22no—
164#125—41no—
165#125—41no—
166#125—41no—
167#125—41no—
168#125—41no—
169#125—41no—
170#125—41no—
171#125—41no—
172#117—82no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C22⋊C4—41no—
213C2×C42—41no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232#103—216no—
233#103—216no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240#97—22no—
241#97—22no—
242#97—22no—
243#97—22no—
244#97—41no—
245#97—41no—
246#97—41no—
247#97C22⋊C414yes—
248C4×C8—41no—
249C4×C8—41no—
250#95—41no—
251#95—41no—
252C24#12111yescharacteristic
253C2×Q8—24no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—84no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—84no—
280C22×C4—82no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12411yescharacteristic
298C22×C4#12711yescharacteristic
299C2×C8—24no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—44no—
308C2×C8—82no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—82no—
336C22⋊C4—82no—
337C42—82no—
338C42C2×C22⋊C411yescharacteristic
339C42C2×C22⋊C411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42#12011yescharacteristic
343C42#12011yescharacteristic
344C42#12011yescharacteristic
345C42#12111yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12311yescharacteristic
352C42#12411yescharacteristic
353C42#12711yescharacteristic
354C23—22no—
355C23—22no—
356C23#38611yescharacteristic
357C23#39211yescharacteristic
358C23#39411yescharacteristic
359C23#47112yes—
360C23#47312yes—
361Q8—48no—
362Q8—84no—
363Q8—84no—
364Q8—84no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#37911yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#39011yescharacteristic
405C2×C4#39011yescharacteristic
406C2×C4#39211yescharacteristic
407C2×C4#39211yescharacteristic
408C2×C4#39211yescharacteristic
409C2×C4#39411yescharacteristic
410C2×C4#39411yescharacteristic
411C2×C4#39411yescharacteristic
412C2×C4#45414yes—
413C8—44no—
414C8—82no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—24no—
420C22—24no—
421C22—21no—
422C22—21no—
423C22#143811yescharacteristic
424C22#190812yes—
425C22#200312yes—
426C22#200912yes—
427C22#201911yescharacteristic
428C22#202011yescharacteristic
429C22#205212yes—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4—162no—
444C4#188812yes—
445C4#200312yes—
446C2—22no—
447C2—22no—
448C2#1013412yes—
449C2#1013512yes—
450C2#1097611yescharacteristic
451C2#1102111yescharacteristic
452C2#1103111yescharacteristic
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
15. C2×#491C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
#218382
#218342
C2×#15562
C2×#14702
C2×#13022
#80342