Finite Group Databasebeta

← #111187

Subgroups — #111187

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1989C211yesmaximal, characteristic
2#20535C211yesmaximal, characteristic
3C2×#1528C211yesmaximal, characteristic
4C2×#1470C211yesmaximal, characteristic
5C2×#1290C211yesmaximal, characteristic
6C2×#1283C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1989—21no—
16#1989—21no—
17#1989—22no—
18#1771—22no—
19#1770—21no—
20C8×C4⋊C4—21no—
21#1749—21no—
22C4×C22⋊C8—21no—
23C2×#340C412yes—
24C2×#340C412yes—
25#1528—22no—
26#1528C414yescomplemented
27C2×#330—22no—
28#1470—22no—
29#1470C414yescomplemented
30C2×#324—21no—
31C2×#324—21no—
32#1290—22no—
33#1283—22no—
34#1279—22no—
35C22×M16—21no—
36C23×C8C2×C412yes—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#432C2×C412yes—
63#432D412yes—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
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⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D411yescharacteristic
80C2×C22⋊C8D411yescharacteristic
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16C2×C412yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97—22no—
88C2×#97D412yes—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4D412yes—
97C2×#95—21no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#336—22no—
103#336—22no—
104#330—28no—
105#324—24no—
106#324—24no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—24no—
114C2×M16—22no—
115C2×M16—42no—
116C2×M16—41no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—44no—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134#125—41no—
135#125—41no—
136#125—41no—
137#125—41no—
138#125—41no—
139#125—41no—
140#125—41no—
141#125—41no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C4212yes—
168C2×C4⋊C4C4212yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—82no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
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—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8C22⋊C412yes—
204C22⋊C8C22⋊C412yes—
205#97—22no—
206#97—22no—
207#97—22no—
208#97—41no—
209#97—41no—
210#97—82no—
211#97C22⋊C414yes—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—41no—
216C4×C8—82no—
217#95—22no—
218#95—21no—
219#95—21no—
220C24C4×D411yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—44no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4C2×C4211yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261M16—24no—
262M16—44no—
263M16—42no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—82no—
281C2×C8—82no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4#10114yes—
303C42—42no—
304C42—42no—
305C42—82no—
306C42C2×C22⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12411yescharacteristic
320C42#12611yescharacteristic
321C42#12711yescharacteristic
322C23—22no—
323C23—21no—
324C23—21no—
325C23C4×C22⋊C411yescharacteristic
326C23#38611yescharacteristic
327C23#38811yescharacteristic
328C23#44312yes—
329C23#44412yes—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4C4×C22⋊C411yescharacteristic
372C2×C4C4×C22⋊C411yescharacteristic
373C2×C4#38611yescharacteristic
374C2×C4#38611yescharacteristic
375C2×C4#38811yescharacteristic
376C2×C4#38811yescharacteristic
377C2×C4#38811yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#39011yescharacteristic
381C2×C4#39011yescharacteristic
382C2×C4#39011yescharacteristic
383C2×C4C2×#10112yes—
384C2×C4#41312yes—
385C8—42no—
386C8—42no—
387C8—44no—
388C8—44no—
389C8—82no—
390C8—82no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22#143711yescharacteristic
400C22#176411yescharacteristic
401C22#176511yescharacteristic
402C22#187612yes—
403C22#190112yes—
404C22#190411yescharacteristic
405C22#190511yescharacteristic
406C22#192912yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4C4×#10112yes—
422C4#196611yescharacteristic
423C4#196711yescharacteristic
424C2—22no—
425C2—21no—
426C2—21no—
427C2#924512yes—
428C2#924612yes—
429C2#1093411yescharacteristic
430C2#1095411yescharacteristic
431C2#1096311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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
26. #1528C4semidirect
29. #1470C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19892
#205352
C2×#15282
C2×#14702
C2×#12902
C2×#12832
#79042