Finite Group Databasebeta

← #96757

Subgroups — #96757

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14699C211yesmaximal, characteristic
2#10795C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1271C212yesmaximal
5#7025C212yesmaximal
6C2×#389C411yescharacteristic
7C2×#389C2211yescharacteristic
8C2×#387—21no—
9C2×C43C2211yescharacteristic
10C2×#336—21no—
11C2×#336—21no—
12C2×#336—21no—
13C2×#336—21no—
14C2×#336C2212yes—
15C2×#336C2212yes—
16#1436—21no—
17#1436—21no—
18C4⋊#95C411yescharacteristic
19C4⋊#95C411yescharacteristic
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1271C418yescomplemented
24C23×C8—22no—
25C23×C8—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39#389—22no—
40#389—42no—
41#389C2×C414yes—
42#387—44no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C43—42no—
54#336—22no—
55#336—22no—
56#336—42no—
57#336—42no—
58#336—42no—
59#336—42no—
60#336C2×C414yes—
61#336C2×C414yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—24no—
74C22×C8—24no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C4212yes—
107C2×C4⋊C4C4212yes—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—21no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42SD1612yes—
129C2×C42Q1612yes—
130C2×C42C22×C412yes—
131C2×C42#4112yes—
132#95—24no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142C24C2×C22⋊C411yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4#9912yes—
208C22×C4#10112yes—
209C22×C4#10312yes—
210C22×C4#10312yes—
211C22×C4C2×C4211yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4C4×D412yes—
214C22×C4C4×D412yes—
215C22×C4#13112yes—
216C22×C4#13112yes—
217C22×C4C2×SD1611yescharacteristic
218C22×C4C2×Q1611yescharacteristic
219C22×C4#13512yes—
220C22×C4AGL(1,ℤ/8)12yes—
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4#13711yescharacteristic
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—84no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—44no—
251C42—44no—
252C42—44no—
253C42—44no—
254C42—42no—
255C42—42no—
256C23C4×C22⋊C411yescharacteristic
257C23#40811yescharacteristic
258C23#40811yescharacteristic
259C23C2×#9911yescharacteristic
260C23#41011yescharacteristic
261C23C2×#10111yescharacteristic
262C23#41311yescharacteristic
263C23C2×#10311yescharacteristic
264C23#41611yescharacteristic
265C23#41711yescharacteristic
266C23#41811yescharacteristic
267C23#42111yescharacteristic
268C23#42111yescharacteristic
269C23#42111yescharacteristic
270C23#42111yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4#40512yes—
352C2×C4#43312yes—
353C2×C4C4×SD1612yes—
354C2×C4C4×Q1612yes—
355C2×C4#44012yes—
356C2×C4#44212yes—
357C8—84no—
358C8—84no—
359C22#146811yescharacteristic
360C22#147011yescharacteristic
361C22#147211yescharacteristic
362C22#147311yescharacteristic
363C22#148111yescharacteristic
364C22#148111yescharacteristic
365C22#148211yescharacteristic
366C22#148211yescharacteristic
367C22#149511yescharacteristic
368C22#149511yescharacteristic
369C22#149611yescharacteristic
370C22#149611yescharacteristic
371C22#151011yescharacteristic
372C22#151211yescharacteristic
373C22#151311yescharacteristic
374C22#151411yescharacteristic
375C22#151611yescharacteristic
376C22#151811yescharacteristic
377C22#152011yescharacteristic
378C22#152111yescharacteristic
379C22#153411yescharacteristic
380C22#153511yescharacteristic
381C22#153511yescharacteristic
382C22#153711yescharacteristic
383C22#154911yescharacteristic
384C22#155011yescharacteristic
385C22#155211yescharacteristic
386C22#155311yescharacteristic
387C22#175011yescharacteristic
388C22C4×#9911yescharacteristic
389C22C4×#10111yescharacteristic
390C22#175911yescharacteristic
391C22#175911yescharacteristic
392C22C4×#10311yescharacteristic
393C22#176211yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—161no—
407C4—161no—
408C2#771511yescharacteristic
409C2#771711yescharacteristic
410C2#771811yescharacteristic
411C2#772011yescharacteristic
412C2#772511yescharacteristic
413C2#772611yescharacteristic
414C2#773111yescharacteristic
415C2#773311yescharacteristic
416C2#815411yescharacteristic
417C2#817911yescharacteristic
418C2#818211yescharacteristic
419C2#821611yescharacteristic
420C2#822111yescharacteristic
421C2#822511yescharacteristic
422C2#823711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1271C4semidirect

Maximal subgroups

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

subgroupindex
#146992
#107952
C2×#12712
#70252