Finite Group Databasebeta

← #99214

Subgroups — #99214

Order 512.

608 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 92 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#13948C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8700C211yesmaximal, characteristic
4#7035C212yesmaximal
5#7035C212yesmaximal
6C2×#388—21no—
7C2×C4×C22⋊C4C2211yescharacteristic
8#1779C2212yes—
9C2×#342—21no—
10C2×#342—21no—
11#1442—21no—
12#1439—21no—
13#1436—21no—
14C4⋊#95C411yescharacteristic, complemented
15C4⋊#95C411yescharacteristic, complemented
16#1434C411yescharacteristic, complemented
17C22⋊#95C2212yes—
18C22⋊#95C2211yescharacteristic
19C22⋊#95C2211yescharacteristic
20C4×#95C411yescharacteristic, complemented
21#1280—21no—
22#1280—21no—
23C24×C4C2311yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37#388—44no—
38C4×C22⋊C4—22no—
39C4×C22⋊C4—22no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95D411yescharacteristic
55#342—24no—
56#342—42no—
57C25C2×D411yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4—42no—
65C23×C4—81no—
66C23×C4—81no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4#4112yes—
74C23×C4#4112yes—
75C22×C8—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—81no—
81C2×C4⋊C4—81no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—81no—
98C2×C22⋊C4—81no—
99C2×C22⋊C4SD1612yes—
100C2×C22⋊C4Q1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—81no—
108C2×C42—81no—
109C2×C42—81no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114#95—22no—
115#95—22no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—81no—
125#95—81no—
126#95—81no—
127#95—82no—
128#95—82no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—41no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24C2×C22⋊C411yescharacteristic
138C24#12312yes—
139C24C2×SD1611yescharacteristic
140C24C2×Q1611yescharacteristic
141C24AGL(1,ℤ/8)12yes—
142C22×C4—21no—
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—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
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—44no—
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—84no—
203C22×C4—84no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4#10312yes—
211C22×C4#10312yes—
212C22×C4#10412yes—
213C22×C4#11711yescharacteristic
214C22×C4#11711yescharacteristic
215C22×C4#12312yes—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—84no—
227C22⋊C4—84no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—81no—
231C42—81no—
232C42—81no—
233C42—81no—
234C42—84no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23#38111yescharacteristic
266C23#41011yescharacteristic
267C23#41011yescharacteristic
268C23C2×#10311yescharacteristic
269C23#41811yescharacteristic
270C23C2×#10411yescharacteristic
271C23#42111yescharacteristic
272C23#48112yes—
273C23#48412yes—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—84no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—162no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4—162no—
344C8—162no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—82no—
368C22—82no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#150311yescharacteristic
374C22#150511yescharacteristic
375C22#151411yescharacteristic
376C22#151811yescharacteristic
377C22#182011yescharacteristic
378C22#182411yescharacteristic
379C22#182611yescharacteristic
380C22#182711yescharacteristic
381C22#182711yescharacteristic
382C22#182811yescharacteristic
383C22#182811yescharacteristic
384C4—42no—
385C4—41no—
386C4—41no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—82no—
392C4—81no—
393C4—161no—
394C4—161no—
395C4—164no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—42no—
402C2#871711yescharacteristic
403C2#874011yescharacteristic
404C2#875011yescharacteristic
405C2#876111yescharacteristic
406C2#876111yescharacteristic
407C2#877611yescharacteristic
408C2#877911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C4⋊#95C4semidirect
15. C4⋊#95C4semidirect
16. #1434C4semidirect
20. C4×#95C4semidirect

Maximal subgroups

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

subgroupindex
#139482
#108012
#87002
#70352