Finite Group Databasebeta

← #110515

Subgroups — #110515

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21314C211yesmaximal, characteristic
2#20480C211yesmaximal, characteristic
3C2×#1528C211yesmaximal, characteristic
4C2×#1470C211yesmaximal, characteristic
5C2×#1307C211yesmaximal, characteristic
6C2×#1299C211yesmaximal, characteristic
7#7902C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1989—21no—
16#1989—21no—
17#1989—22no—
18#1770—21no—
19C8×C4⋊C4—21no—
20#1753—22no—
21#1745—21no—
22#1736C412yes—
23#1736C412yes—
24C4×#97—21no—
25#1528—22no—
26#1528C414yescomplemented
27C2×#330—21no—
28C2×#330—21no—
29#1470—22no—
30#1470C414yescomplemented
31C2×#324—22no—
32#1307—22no—
33#1299—22no—
34#1279—22no—
35C22×M16—21no—
36C22×M16C2×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—
49C2×#117C2×C412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16C2×C412yes—
82C4×M16D412yes—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—22no—
86C2×#97D411yescharacteristic
87C2×#97D411yescharacteristic
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D412yes—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96#376—21no—
97C43—21no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#336—22no—
103#336—22no—
104#330—24no—
105#330—24no—
106#324—28no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—44no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—41no—
134#125—41no—
135#125—41no—
136#125—41no—
137#125—41no—
138#125—41no—
139#125—41no—
140#125—41no—
141#117—24no—
142#117—42no—
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—21no—
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—41no—
158C2×C4⋊C4—41no—
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⋊C4C4212yes—
166C2×C4⋊C4C4212yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
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
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201C22⋊C8C22⋊C414yes—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—21no—
206#97—21no—
207#97—44no—
208#97—41no—
209#97C22⋊C412yes—
210#97C22⋊C412yes—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—44no—
215C4×C8—41no—
216#95—22no—
217#95—21no—
218#95—21no—
219C24C4×D411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—81no—
235C22×C4—81no—
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#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254M16—44no—
255M16—44no—
256M16—82no—
257M16—82no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—44no—
271C2×C8—82no—
272C2×C8—81no—
273C2×C8—81no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4#10114yes—
294C22⋊C4—44no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—82no—
299C42C2×C22⋊C411yescharacteristic
300C42#11711yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12311yescharacteristic
311C42#12411yescharacteristic
312C42#12411yescharacteristic
313C42#12611yescharacteristic
314C42#12711yescharacteristic
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23C4×C22⋊C411yescharacteristic
320C23#38611yescharacteristic
321C23#38811yescharacteristic
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4C4×C22⋊C411yescharacteristic
359C2×C4C4×C22⋊C411yescharacteristic
360C2×C4#38611yescharacteristic
361C2×C4#38611yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#39011yescharacteristic
368C2×C4#39011yescharacteristic
369C2×C4#39011yescharacteristic
370C2×C4C2×#10112yes—
371C2×C4#41312yes—
372C8—24no—
373C8—44no—
374C8—42no—
375C8—82no—
376C8—82no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22#143711yescharacteristic
387C22#188112yes—
388C22#190112yes—
389C22#190611yescharacteristic
390C22#190611yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—44no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4C4×#10112yes—
406C4#196611yescharacteristic
407C4#196711yescharacteristic
408C2—41no—
409C2—41no—
410C2#935112yes—
411C2#935212yes—
412C2#1094511yescharacteristic
413C2#1095611yescharacteristic
414C2#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
30. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#213142
#204802
C2×#15282
C2×#14702
C2×#13072
C2×#12992
#79022