Finite Group Databasebeta

← #99164

Subgroups — #99164

Order 512.

690 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 124 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#13802C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7028C212yesmaximal, complemented
5#7024C212yesmaximal, complemented
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9C22×#95C411yescharacteristic
10#1797C2212yes—
11C4×C22⋊C8C2212yes—
12C2×#323—21no—
13C2×#323—21no—
14#1436—21no—
15#1436—21no—
16#1434C411yescharacteristic
17#1434C2211yescharacteristic
18C22⋊#95—21no—
19C4×#95C411yescharacteristic
20C4×#95C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C4×C8—22no—
37#380—22no—
38#380—22no—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C4×C22⋊C4—42no—
42C4×C22⋊C4—42no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95C814yescomplemented
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58#336—22no—
59#336—22no—
60#323—24no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—24no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C2×C812yes—
84C23×C4M1612yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—42no—
89C22×C8—42no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4D812yes—
102C2×C22⋊C4SD1612yes—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42—81no—
113C2×C42—81no—
114C2×C42#4112yes—
115C2×C42#4112yes—
116C4⋊C8—82no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C4×C8—82no—
122#95—24no—
123#95—22no—
124#95—22no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—81no—
135#95—81no—
136#95C2×C812yes—
137#95C2×C812yes—
138#95M1612yes—
139#95M1612yes—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24C22⋊C812yes—
149C24#10212yes—
150C24#10412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—48no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
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—48no—
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—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C22⋊C812yes—
221C22×C4#10212yes—
222C22×C4#11711yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4#12312yes—
225C22×C4#12312yes—
226C22×C4C22×C811yescharacteristic
227C22×C4C2×M1611yescharacteristic
228C22×C4C2×M1611yescharacteristic
229C22×C4C2×M1611yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×SD1611yescharacteristic
232C22×C4#13712yes—
233C2×C8—44no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C4⋊C4—84no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—44no—
243C22⋊C4—48no—
244C42—42no—
245C42—42no—
246C42—81no—
247C42—81no—
248C42—82no—
249C42—82no—
250C42—84no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23D4⋊C814yes—
278C23#38111yescharacteristic
279C23C2×C22⋊C811yescharacteristic
280C23#40711yescharacteristic
281C23C2×#10211yescharacteristic
282C23#41911yescharacteristic
283C23C2×#10411yescharacteristic
284C23#42111yescharacteristic
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—816no—
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#43111yescharacteristic
352C2×C4#43111yescharacteristic
353C2×C4#43211yescharacteristic
354C2×C4#43211yescharacteristic
355C2×C4#48012yes—
356C2×C4#48312yes—
357C8—84no—
358C8—162no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22C2×D4⋊C811yescharacteristic
383C22#147911yescharacteristic
384C22#148611yescharacteristic
385C22#149111yescharacteristic
386C22#181711yescharacteristic
387C22#182311yescharacteristic
388C22#182611yescharacteristic
389C22#182711yescharacteristic
390C22#182711yescharacteristic
391C22#182811yescharacteristic
392C22#182911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—44no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—81no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—44no—
412C2#870611yescharacteristic
413C2#873011yescharacteristic
414C2#873411yescharacteristic
415C2#876011yescharacteristic
416C2#876411yescharacteristic
417C2#877211yescharacteristic
418C2#877611yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7028C2semidirect
5. #7024C2semidirect
51. C2×#95C8semidirect

Maximal subgroups

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

subgroupindex
#138022
#85572
C4×#3232
#70282
#70242