Finite Group Databasebeta

← #110537

Subgroups — #110537

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21332C211yesmaximal, characteristic
2#20546C211yesmaximal, characteristic
3C2×#1552C211yesmaximal, characteristic
4#17676C211yesmaximal, characteristic
5#8044C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7930C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1996—22no—
16#1996—21no—
17#1996—21no—
18#1771—22no—
19#1767—21no—
20C4×C4⋊C8—21no—
21#1752—21no—
22C8×C22⋊C4—21no—
23#1740C414yes—
24#1552—22no—
25#1552—21no—
26#1552—21no—
27C2×#332C412yescomplemented
28C2×#332C412yescomplemented
29#1534C414yescomplemented
30#1531—22no—
31#1531—22no—
32#1296—22no—
33#1285—22no—
34#1282—22no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117C2×C412yes—
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#432C2×C412yes—
66#432D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8C2×C412yes—
74C2×C4⋊C8D412yes—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—22no—
83C2×#97D412yes—
84C2×C4×C8—22no—
85#384—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4D412yes—
94C4×C22⋊C4D411yescharacteristic
95C4×C22⋊C4D411yescharacteristic
96#376—21no—
97C43—21no—
98#342—24no—
99#342—24no—
100#336—22no—
101#336—22no—
102#332—22no—
103#332—22no—
104#332—22no—
105#332—22no—
106#330—28no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—82no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
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—24no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—42no—
154#117—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4C4212yes—
164C2×C4⋊C4C4212yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—41no—
197C4⋊C8C22⋊C412yes—
198C4⋊C8C22⋊C412yes—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205#97—22no—
206#97—22no—
207#97—41no—
208#97—41no—
209#97—41no—
210#97—82no—
211#97C22⋊C414yes—
212C4×C8—82no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—41no—
217#95—41no—
218C24C2×C22⋊C411yescharacteristic
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—81no—
224C2×Q8—81no—
225C2×Q8—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C2×C4211yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12711yescharacteristic
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—82no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—44no—
290C22⋊C4#10114yes—
291C42—44no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—82no—
300C42C2×C22⋊C411yescharacteristic
301C42C2×C22⋊C411yescharacteristic
302C42#11711yescharacteristic
303C42#11711yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12311yescharacteristic
311C42#12311yescharacteristic
312C42#12411yescharacteristic
313C42#12411yescharacteristic
314C42#12611yescharacteristic
315C42#12611yescharacteristic
316C23—41no—
317C23—41no—
318C23C4×C22⋊C411yescharacteristic
319C23#38611yescharacteristic
320C23#39011yescharacteristic
321C23C2×#10112yes—
322C23#41312yes—
323Q8—82no—
324Q8—82no—
325Q8—81no—
326Q8—81no—
327Q8—82no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#38811yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#39011yescharacteristic
375C2×C4#39011yescharacteristic
376C8—44no—
377C8—42no—
378C8—42no—
379C8—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22#143711yescharacteristic
390C22C4×#10112yes—
391C22#190112yes—
392C22#196611yescharacteristic
393C22#196711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4#190411yescharacteristic
410C4#190511yescharacteristic
411C4#197612yes—
412C2—22no—
413C2—21no—
414C2—21no—
415C2—22no—
416C2#1094411yescharacteristic
417C2#1095411yescharacteristic
418C2#1096311yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. C2×#332C4semidirect
28. C2×#332C4semidirect
29. #1534C4semidirect

Maximal subgroups

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

subgroupindex
#213322
#205462
C2×#15522
#176762
#80442
#79522
#79302