Finite Group Databasebeta

← #118098

Subgroups — #118098

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21870C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4C2×#1537C211yesmaximal, characteristic
5C2×#1287C211yesmaximal, characteristic
6#7912C211yesmaximal, characteristic
7#7900C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993—21no—
16#1987—21no—
17#1981—21no—
18#1960—21no—
19#1948—21no—
20#1947—21no—
21#1873—22no—
22#1872—21no—
23#1872—21no—
24#1853—21no—
25#1852—21no—
26#1803—21no—
27#1803—21no—
28#1776—22no—
29C4×#103—22no—
30#1537—22no—
31#1537C414yescomplemented
32C2×#330—22no—
33#1287—22no—
34#1280—22no—
35#1279—22no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67#432C2×C412yes—
68#432D412yes—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73C2×#107—22no—
74C2×#106—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8D412yes—
87C2×#97—21no—
88C2×#97—21no—
89C2×#97D412yes—
90C2×C4×C8D412yes—
91#384D411yescharacteristic
92#384D411yescharacteristic
93#382—21no—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C2×#95—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#330—28no—
110C22×Q8—21no—
111C22×Q8—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153#117—82no—
154C2×C4⋊C4—21no—
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—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—82no—
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—44no—
192#106—44no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201#103—44no—
202#103—44no—
203#103—82no—
204#103—82no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#97—22no—
214#97—22no—
215#97—41no—
216#97—41no—
217#97—41no—
218#97C22⋊C414yes—
219C4×C8—22no—
220C4×C8—22no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225C24#12111yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4#11711yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12611yescharacteristic
272C22×C4#12611yescharacteristic
273C22×C4#12711yescharacteristic
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—
284C2×C8—81no—
285C2×C8—81no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—82no—
315C42—42no—
316C42—42no—
317C42—82no—
318C42—82no—
319C42—82no—
320C42C2×C22⋊C411yescharacteristic
321C42C2×C4⋊C411yescharacteristic
322C42C4×D411yescharacteristic
323C42C4×D411yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12411yescharacteristic
335C42#12411yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#38811yescharacteristic
343C23#39211yescharacteristic
344C23#39611yescharacteristic
345Q8—44no—
346Q8—42no—
347Q8—42no—
348Q8—82no—
349Q8—81no—
350Q8—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#38011yescharacteristic
396C2×C4#38511yescharacteristic
397C2×C4#38511yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#38811yescharacteristic
400C2×C4#39011yescharacteristic
401C2×C4#39311yescharacteristic
402C2×C4#39411yescharacteristic
403C2×C4#39411yescharacteristic
404C2×C4#39611yescharacteristic
405C2×C4#39711yescharacteristic
406C2×C4#39911yescharacteristic
407C2×C4#47112yes—
408C2×C4#47312yes—
409C8—44no—
410C8—82no—
411C8—82no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#143911yescharacteristic
424C22#200811yescharacteristic
425C22#200911yescharacteristic
426C22#201011yescharacteristic
427C22#201011yescharacteristic
428C22#201911yescharacteristic
429C22#202011yescharacteristic
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—82no—
443C4—82no—
444C4—161no—
445C4—161no—
446C4#193012yes—
447C4#206412yes—
448C2—41no—
449C2—41no—
450C2#1003911yescharacteristic
451C2#1004111yescharacteristic
452C2#1004311yescharacteristic
453C2#1004511yescharacteristic
454C2#1108311yescharacteristic
455C2#1111711yescharacteristic
456C2#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
31. #1537C4semidirect

Maximal subgroups

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

subgroupindex
#218702
#213852
#213652
C2×#15372
C2×#12872
#79122
#79002