Finite Group Databasebeta

← #109713

Subgroups — #109713

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2#20525C211yesmaximal, characteristic
3#8057C211yesmaximal, characteristic
4C2×#1298C211yesmaximal, characteristic
5C2×#1298C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855C414yescomplemented
17#1810—22no—
18#1753—22no—
19#1752—21no—
20C8×C22⋊C4—21no—
21#1749—21no—
22C4×C22⋊C8—21no—
23#1298—22no—
24#1298—21no—
25#1298—21no—
26#1298C414yescomplemented
27#1282—21no—
28#1282—21no—
29#1282C414yes—
30C24×C4C2311yescharacteristic
31C22×M16—21no—
32C22×M16C2×C412yes—
33C22×M16D412yes—
34C23×C8—21no—
35C23×C8—21no—
36C23×C8—21no—
37C22×C22⋊C4Q812yes—
38C22×C42—21no—
39C2×#108—24no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407C2×C412yes—
50#407C2×C412yes—
51#407D412yes—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8C2×C412yes—
70C2×C22⋊C8C2×C412yes—
71C2×C22⋊C8C2×C412yes—
72C2×C22⋊C8C2×C412yes—
73C2×C22⋊C8C2×C412yes—
74C2×C22⋊C8D411yescharacteristic
75C2×C22⋊C8D411yescharacteristic
76C2×C22⋊C8Q812yes—
77C2×#97—22no—
78C2×C4×C8—22no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4C2×C412yes—
82C4×C22⋊C4D411yescharacteristic
83C4×C22⋊C4D411yescharacteristic
84C2×#95—21no—
85#341—24no—
86#336—22no—
87#336—22no—
88C25C2×Q811yescharacteristic
89C23×C4—22no—
90C23×C4—82no—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—82no—
111C2×M16—82no—
112C2×M16—82no—
113C2×M16C22⋊C414yes—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—82no—
128C22×C8—82no—
129C22×C8—82no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4C4⋊C412yes—
134C2×C22⋊C4C4⋊C412yes—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—22no—
138C2×C42—82no—
139#108—84no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8C4214yes—
164C22⋊C8C4⋊C412yes—
165C22⋊C8C4⋊C412yes—
166#97—82no—
167C4×C8—82no—
168#95—22no—
169#95—21no—
170#95—21no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24—81no—
175C24C2×C4⋊C411yescharacteristic
176C24C4×Q811yescharacteristic
177C24C4×Q811yescharacteristic
178C24C22⋊Q811yescharacteristic
179C24C22⋊Q811yescharacteristic
180C24#12511yescharacteristic
181C24C4⋊Q811yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—82no—
201C22×C4C2×C4211yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C4⋊C411yescharacteristic
204C22×C4#11711yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×Q811yescharacteristic
216C22×C4C4×Q811yescharacteristic
217C22×C4#12011yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12511yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12611yescharacteristic
229M16—44no—
230M16—44no—
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235C2×C8—24no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—44no—
260C22⋊C4#10814yescomplemented
261C42—42no—
262C42—42no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—84no—
287C23—84no—
288C23C4×C4⋊C411yescharacteristic
289C23#38511yescharacteristic
290C23#38711yescharacteristic
291C23#38711yescharacteristic
292C23#38911yescharacteristic
293C23#38911yescharacteristic
294C23#39111yescharacteristic
295C23C2×#10812yes—
296C23#43012yes—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—41no—
306C2×C4—41no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—88no—
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4C4×C4⋊C411yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38711yescharacteristic
337C2×C4#38711yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#38811yescharacteristic
340C8—44no—
341C8—42no—
342C8—42no—
343C8—44no—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—82no—
372C22—81no—
373C22—81no—
374C22#143611yescharacteristic
375C22C4×#10811yescharacteristic
376C22#177811yescharacteristic
377C22#185512yes—
378C22#195012yes—
379C22#195312yes—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—44no—
388C4—82no—
389C4#191312yes—
390C2—22no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—41no—
398C2#1089811yescharacteristic
399C2#1091111yescharacteristic
400C2#1091511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1436
prime radicalC25C2×Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1855C4semidirect
26. #1298C4semidirect
260. C22⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
#219482
#205252
#80572
C2×#12982
#79292
C2×#12822