Finite Group Databasebeta

← #99565

Subgroups — #99565

Order 512.

698 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 108 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#21916C211yesmaximal, characteristic
2C4×#395C211yesmaximal, characteristic
3#8311C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7028C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C2×#395C411yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1917—24no—
11#1817—21no—
12#1797C2212yes—
13#1749—21no—
14C4×C22⋊C8C2212yes—
15C2×Q8⋊C8—24no—
16C2×#324—21no—
17#1446C411yescharacteristic
18#1444—21no—
19#1444C411yescharacteristic
20#1441—21no—
21#1441C411yescharacteristic
22C4×#95—21no—
23C4×#95C2211yescharacteristic
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C2×C4×Q8—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8D414yes—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44C2×C4×C8—22no—
45#395—22no—
46#395—22no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4—44no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#336—24no—
62#336—22no—
63#336—22no—
64#336—22no—
65Q8⋊C8—48no—
66#324—24no—
67C22×Q8—21no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8M1612yes—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—42no—
86C4×Q8—28no—
87C4×Q8—44no—
88C4×Q8—44no—
89C4×Q8—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42C2×D412yes—
124C2×C42#4112yes—
125C4⋊C8—44no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130#97—82no—
131C4×C8—44no—
132#95—24no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—22no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95M1612yes—
144C24C2×C22⋊C411yescharacteristic
145C2×Q8—24no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—44no—
152C2×Q8—48no—
153C2×Q8—42no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
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—41no—
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—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4#10412yes—
212C22×C4C4×D412yes—
213C22×C4#12011yescharacteristic
214C22×C4#12112yes—
215C22×C4#12311yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C22×C4C2×M1611yescharacteristic
218C22×C4#13112yes—
219C22×C4#13512yes—
220C22×C4#13712yes—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—82no—
226C2×C8—82no—
227C4⋊C4—24no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—42no—
231C4⋊C4—48no—
232C4⋊C4—48no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C42—24no—
236C42—42no—
237C42—48no—
238C42—44no—
239C42—42no—
240C42—42no—
241C42—44no—
242C42—42no—
243C42—81no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23#38611yescharacteristic
249C23#40711yescharacteristic
250C23#40811yescharacteristic
251C23#41611yescharacteristic
252C23#41911yescharacteristic
253C23C2×#10411yescharacteristic
254C23#42111yescharacteristic
255Q8—28no—
256Q8—44no—
257Q8—44no—
258Q8—44no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
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#43512yes—
324C2×C4#43612yes—
325C2×C4#45212yes—
326C2×C4#45612yes—
327C2×C4B(4,𝔽2)12yes—
328C2×C4#48512yes—
329C8—84no—
330C8—84no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22#148811yescharacteristic
344C22#149111yescharacteristic
345C22#149411yescharacteristic
346C22#149711yescharacteristic
347C22#187011yescharacteristic
348C22#187111yescharacteristic
349C22#187311yescharacteristic
350C22#188411yescharacteristic
351C22#188611yescharacteristic
352C22#188911yescharacteristic
353C22#189111yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—44no—
358C4—44no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4#160314yes—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2#894111yescharacteristic
372C2#896211yescharacteristic
373C2#896711yescharacteristic
374C2#901011yescharacteristic
375C2#901411yescharacteristic
376C2#903011yescharacteristic
377C2#903411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#219162
C4×#3952
#83112
C4×#3242
#70282