Finite Group Databasebeta

← #106643

Subgroups — #106643

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2346C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#8072C212yesmaximal, complemented
5#7959C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9#2346—24no—
10#2346—24no—
11C2×#389—21no—
12C2×#388—21no—
13C2×#388—21no—
14C2×#382C411yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#380C411yescharacteristic
17C2×C4×C4⋊C4—21no—
18C2×C4×C22⋊C4C411yescharacteristic
19C2×#376C411yescharacteristic
20C2×#376C2211yescharacteristic
21#1464—22no—
22#1464C414yescomplemented
23#1463—21no—
24#1463—21no—
25#1462—21no—
26#1462—21no—
27#1458—22no—
28#1458C414yescomplemented
29#1300—24no—
30#1286—24no—
31C24×C4C2×C411yescharacteristic
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117D412yes—
39C2×#117Q812yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2311yescharacteristic
54#432—22no—
55#432—22no—
56#432—22no—
57#432C2×C414yes—
58#431—22no—
59#431—22no—
60#431—22no—
61#431C2×C414yes—
62#423—22no—
63#423—22no—
64#407—22no—
65#407—22no—
66C4×M16—22no—
67C4×M16—22no—
68#389—28no—
69#388—28no—
70#388—28no—
71#382—22no—
72#382—22no—
73#382—22no—
74#382D414yes—
75#380—22no—
76#380—22no—
77C4×C4⋊C4—28no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80#376—22no—
81#376—22no—
82#376—22no—
83#376D414yes—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92#324—24no—
93#324—24no—
94#323—24no—
95#323—24no—
96C25C22⋊C411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4C4212yes—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C4⋊C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×M16—42no—
115C2×M16—42no—
116#117—24no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—24no—
121#117—24no—
122#117—24no—
123#117—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C4⋊C412yes—
137C2×C4⋊C4C4⋊C412yes—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—82no—
146C2×C22⋊C4C22⋊C412yes—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C4⋊C412yes—
149C2×C22⋊C4C4⋊C412yes—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—44no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42C4212yes—
161C2×C42C4⋊C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42#4112yes—
169C2×C42#4112yes—
170C4⋊C8—24no—
171C4⋊C8—24no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C22⋊C8—24no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178#97—24no—
179#97—42no—
180C4×C8—24no—
181C4×C8—42no—
182#95—24no—
183#95—24no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—24no—
189#95—24no—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194C24—21no—
195C24—21no—
196C24—42no—
197C24—42no—
198C24—41no—
199C24—41no—
200C24#9512yes—
201C24C2×C22⋊C411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4#9512yes—
245C22×C4#9512yes—
246C22×C4#9512yes—
247C22×C4C2×C4211yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4#11711yescharacteristic
256C22×C4#11711yescharacteristic
257C22×C4#12012yes—
258C22×C4#12012yes—
259C22×C4#12312yes—
260C22×C4#12312yes—
261M16—82no—
262M16—82no—
263M16—84no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—48no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—48no—
278C22⋊C4—84no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—48no—
284C42—48no—
285C42—48no—
286C42C4×D412yes—
287C42C4×D412yes—
288C42C4×D412yes—
289C42C4×D412yes—
290C42#12112yes—
291C42#12112yes—
292C42C22⋊Q812yes—
293C42C22⋊Q812yes—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—84no—
311C23—84no—
312C23C2×#9511yescharacteristic
313C23#37911yescharacteristic
314C23#38111yescharacteristic
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4#34414yes—
364C2×C4C4×C22⋊C411yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#38011yescharacteristic
367C2×C4#38011yescharacteristic
368C2×C4#38512yes—
369C2×C4#38512yes—
370C2×C4#38612yes—
371C2×C4#38612yes—
372C8—84no—
373C8—84no—
374C22—24no—
375C22—24no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—84no—
390C22C22⋊#9511yescharacteristic
391C22C2×#34412yes—
392C22#179511yescharacteristic
393C22#179611yescharacteristic
394C22#182712yes—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—82no—
408C4—82no—
409C4—82no—
410C2—21no—
411C2—21no—
412C2—22no—
413C2—22no—
414C2—82no—
415C2#1073811yescharacteristic
416C2#1075811yescharacteristic
417C2#1076511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊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
4. #8072C2semidirect
5. #7959C2semidirect
22. #1464C4semidirect
28. #1458C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
#173572
#80722
#79592