Finite Group Databasebeta

← #111189

Subgroups — #111189

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1986C211yesmaximal, characteristic
2#17715C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#17327C211yesmaximal, characteristic
5#7950C211yesmaximal, characteristic
6#7908C211yesmaximal, characteristic
7#7888C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1986—21no—
16#1986—21no—
17#1986—22no—
18C2×#339C414yes—
19#1552—22no—
20#1549C412yescomplemented
21#1546C412yescomplemented
22#1544—22no—
23#1472—22no—
24#1470C414yescomplemented
25#1461—21no—
26#1461—21no—
27#1453—22no—
28#1451—21no—
29#1451—21no—
30C8×M16—21no—
31C8×M16—21no—
32#1285—22no—
33#1280—22no—
34#1278—22no—
35C22×M16C2×C412yes—
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—
52C22×C4⋊C4—21no—
53C22×C42C2×C412yes—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#431C2×C412yes—
69#431D412yes—
70#431D411yescharacteristic
71#431D411yescharacteristic
72#423—21no—
73#423C2×C412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16D411yescharacteristic
85C4×M16D411yescharacteristic
86C4×M16D412yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C2×#95—21no—
96#339—24no—
97#339—24no—
98#332—24no—
99#332—24no—
100#324—24no—
101#324—24no—
102#322—22no—
103#322—22no—
104#322—22no—
105C82—22no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—22no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C2×M16—42no—
116C2×M16—41no—
117C2×M16C22⋊C412yes—
118C2×M16C22⋊C412yes—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C4212yes—
163C2×C4⋊C4C4212yes—
164C2×C22⋊C4—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—44no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8C22⋊C414yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C412yes—
198C22⋊C8C22⋊C412yes—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—21no—
203#97—21no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214#95—41no—
215#95—41no—
216C24C4×D411yescharacteristic
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C2×C4211yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12711yescharacteristic
254M16—22no—
255M16—22no—
256M16—22no—
257M16—44no—
258M16—42no—
259M16—42no—
260M16—82no—
261M16#10114yes—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—82no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C42—42no—
289C42—42no—
290C42—44no—
291C42—82no—
292C42C2×C22⋊C411yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12011yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12411yescharacteristic
307C42#12611yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23C4×C22⋊C411yescharacteristic
314C23#38811yescharacteristic
315C23#39011yescharacteristic
316Q8—82no—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4C4×C22⋊C411yescharacteristic
349C2×C4C4×C22⋊C411yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#39011yescharacteristic
359C2×C4#39011yescharacteristic
360C2×C4C2×#10112yes—
361C2×C4#41312yes—
362C2×C4#44312yes—
363C2×C4#44412yes—
364C8—24no—
365C8—42no—
366C8—42no—
367C8—44no—
368C8—42no—
369C8—82no—
370C8—82no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22#143711yescharacteristic
383C22#176411yescharacteristic
384C22#176511yescharacteristic
385C22#196611yescharacteristic
386C22#196711yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—84no—
396C4—82no—
397C4—82no—
398C4C4×#10112yes—
399C4#187612yes—
400C4#190112yes—
401C4#192912yes—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—41no—
407C2#1093411yescharacteristic
408C2#1095111yescharacteristic
409C2#1096311yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1549C4semidirect
21. #1546C4semidirect
24. #1470C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19862
#177152
#173852
#173272
#79502
#79082
#78882