Finite Group Databasebeta

← #118503

Subgroups — #118503

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2C2×#1573C211yesmaximal, characteristic
3C2×#1552C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#10850C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993—21no—
16#1960—21no—
17#1872—22no—
18#1805—21no—
19#1805—21no—
20C2×#340—22no—
21#1573—22no—
22#1573—22no—
23C2×#335—21no—
24C2×#334—21no—
25#1552—22no—
26#1552—22no—
27C2×#332—21no—
28C2×#332—21no—
29#1503—22no—
30#1503C414yescomplemented
31C2×#328—22no—
32#1285—22no—
33#1280—22no—
34C23×C8—21no—
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—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D412yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8D412yes—
82C2×#97—21no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96#340—22no—
97#340—22no—
98#340—44no—
99#335—24no—
100#334—24no—
101#332—24no—
102#332—24no—
103#328—28no—
104C22×Q8—21no—
105C22×Q8—21no—
106C23×C4—22no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C22×C8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
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—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C42—22no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—82no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197#103—82no—
198#103—82no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8C22⋊C414yes—
205#97—22no—
206#97—41no—
207#97—41no—
208#97—41no—
209C4×C8—22no—
210C4×C8—22no—
211#95—22no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217C24#12111yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—82no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12711yescharacteristic
264C2×C8—24no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—42no—
301C42—42no—
302C42—82no—
303C42—82no—
304C42C2×C4⋊C411yescharacteristic
305C42#11711yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12011yescharacteristic
309C42#12011yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12611yescharacteristic
320C23—22no—
321C23—21no—
322C23—21no—
323C23#39011yescharacteristic
324C23#39411yescharacteristic
325C23#39611yescharacteristic
326C23#47112yes—
327C23#47312yes—
328Q8—42no—
329Q8—42no—
330Q8—82no—
331Q8—82no—
332Q8—81no—
333Q8—81no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—162no—
372C2×C4—162no—
373C2×C4#38011yescharacteristic
374C2×C4#38511yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#38811yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#39211yescharacteristic
380C2×C4#39311yescharacteristic
381C2×C4#39411yescharacteristic
382C2×C4#39611yescharacteristic
383C2×C4#39711yescharacteristic
384C2×C4#39911yescharacteristic
385C2×C4#45311yescharacteristic
386C2×C4#45411yescharacteristic
387C2×C4#45511yescharacteristic
388C2×C4#45611yescharacteristic
389C8—44no—
390C8—82no—
391C8—81no—
392C8—82no—
393C8—81no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22#143911yescharacteristic
403C22#197711yescharacteristic
404C22#197911yescharacteristic
405C22#203711yescharacteristic
406C22#203811yescharacteristic
407C22#205212yes—
408C22#205312yes—
409C22#206412yes—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—84no—
421C4—82no—
422C4—82no—
423C4—162no—
424C4#185911yescharacteristic
425C4#186011yescharacteristic
426C4#200611yescharacteristic
427C4#200711yescharacteristic
428C2—21no—
429C2—22no—
430C2—21no—
431C2#1051012yes—
432C2#1051112yes—
433C2#1109211yescharacteristic
434C2#1111311yescharacteristic
435C2#1112811yescharacteristic
subgroupG/H
center, socleC23#394
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
30. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#218662
C2×#15732
C2×#15522
C2×#15032
#108502
C2×#12852
#79142