Finite Group Databasebeta

← #142168

Subgroups — #142168

Order 512.

884 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15178C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×C22⋊Q8—21no—
7C2×#391—21no—
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1444—21no—
15#1444C412yescomplemented
16#1439—21no—
17#1439C2212yes—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C2×C22⋊Q8—44no—
26C2×C22⋊Q8—44no—
27C2×C22⋊Q8—42no—
28C2×C22⋊Q8—42no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C412yes—
33C22×C22⋊C4—22no—
34C22×C22⋊C4—22no—
35C22×C22⋊C4D411yescharacteristic
36C22×C22⋊C4C2311yescharacteristic
37C22×C42—41no—
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C4⋊C8—41no—
41C2×C22⋊C8—22no—
42#391—44no—
43C4×C22⋊C4—24no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—42no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—41no—
51C2×#95C2×C412yes—
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54#342—24no—
55#342—22no—
56#342Q814yescomplemented
57#336—41no—
58#323—24no—
59C25C2×D411yescharacteristic
60C22×Q8—24no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—42no—
67C22×Q8—42no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4D812yes—
81C23×C4SD1612yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—42no—
85C4⋊Q8—44no—
86C4⋊Q8—48no—
87C4⋊Q8—44no—
88C22⋊Q8—416no—
89C22⋊Q8—416no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4C2×Q812yes—
116C2×C22⋊C4#4112yes—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—44no—
124C2×C42—82no—
125C2×C42—82no—
126C4⋊C8—82no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—44no—
133#95—44no—
134#95—44no—
135#95—42no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24C2×C22⋊C411yescharacteristic
145C24C22⋊Q812yes—
146C24C2×D811yescharacteristic
147C24C2×SD1611yescharacteristic
148C24#13712yes—
149C2×Q8—48no—
150C2×Q8—44no—
151C2×Q8—48no—
152C2×Q8—44no—
153C2×Q8—82no—
154C2×Q8—84no—
155C2×Q8—84no—
156C2×Q8—82no—
157C2×Q8—82no—
158C2×Q8—82no—
159C22×C4—22no—
160C22×C4—22no—
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—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
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—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—84no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4#9911yescharacteristic
215C22×C4#9911yescharacteristic
216C22×C4#10212yes—
217C22×C4#10212yes—
218C22×C4#10412yes—
219C22×C4#10412yes—
220C22×C4C4×Q812yes—
221C22×C4#12411yescharacteristic
222C22×C4C4⋊Q811yescharacteristic
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—416no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C22⋊C4—28no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—81no—
244C42—88no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23#39111yescharacteristic
276C23C2×#9911yescharacteristic
277C23#41011yescharacteristic
278C23C2×#10211yescharacteristic
279C23#41911yescharacteristic
280C23C2×#10411yescharacteristic
281C23C2×#10411yescharacteristic
282C23#47412yes—
283C23#47512yes—
284Q8—88no—
285Q8—88no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4#32712yes—
343C2×C4#32712yes—
344C2×C4AΣL(1,𝔽16)12yes—
345C2×C4#35312yes—
346C8—162no—
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—22no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—82no—
374C22—82no—
375C22C2×#32711yescharacteristic
376C22#150211yescharacteristic
377C22#150611yescharacteristic
378C22#150811yescharacteristic
379C22#198811yescharacteristic
380C22#199411yescharacteristic
381C22#199611yescharacteristic
382C22C2×AΣL(1,𝔽16)11yescharacteristic
383C22#212311yescharacteristic
384C22C2×#35311yescharacteristic
385C22#212911yescharacteristic
386C4—22no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—41no—
391C4—41no—
392C4—84no—
393C4—88no—
394C4—81no—
395C4—84no—
396C4#134314yes—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2#990911yescharacteristic
404C2C2×#134311yescharacteristic
405C2#1169411yescharacteristic
406C2#1178511yescharacteristic
407C2#1178811yescharacteristic
408C2#1329611yescharacteristic
409C2#1329811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Q8C4semidirect
15. #1444C4semidirect
18. C4⋊#95C4semidirect
56. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#151782
C4×#3422
#85572
#72502