Finite Group Databasebeta

← #140669

Subgroups — #140669

Order 512.

516 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 94 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#14003C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3C2×#1319C212yesmaximal
4#8558C211yesmaximal, characteristic
5#8153C211yesmaximal, characteristic
6C2×#391C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×#382C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12#1749—21no—
13C2×#324C2212yes—
14C2×#324C2212yes—
15#1443—21no—
16#1443—21no—
17C4⋊#95C411yescharacteristic
18#1434—21no—
19#1434—21no—
20C4×#95C411yescharacteristic
21#1324—24no—
22#1319—24no—
23#1280—21no—
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×#97—41no—
38#391—22no—
39#391—22no—
40#391—22no—
41#391—22no—
42#391—41no—
43#391—41no—
44#382—22no—
45#382—22no—
46C4×C4⋊C4—41no—
47C4×C4⋊C4—41no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#336—41no—
63#336—41no—
64#324—22no—
65#324—22no—
66#324—22no—
67#324D414yes—
68C22×Q8—21no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8M1612yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—42no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C2×D412yes—
99C2×C4⋊C4#4112yes—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42M1612yes—
117C4⋊C8—82no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—82no—
123#95—24no—
124#95—24no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—24no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—82no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—82no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
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—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#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4C4×D412yes—
213C22×C4#12112yes—
214C22×C4#12411yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C22×C4C2×M1611yescharacteristic
218C22×C4#13112yes—
219C2×C8—44no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C42—24no—
231C42—42no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—42no—
236C42—81no—
237C42—81no—
238C42—82no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23#39011yescharacteristic
244C23#40711yescharacteristic
245C23#40811yescharacteristic
246C23C2×#9911yescharacteristic
247C23#41011yescharacteristic
248C23#41311yescharacteristic
249C23#41311yescharacteristic
250Q8—84no—
251Q8—84no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
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—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
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#35412yes—
323C2×C4#43512yes—
324C2×C4#43512yes—
325C8—84no—
326C8—162no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22#146411yescharacteristic
340C22#146511yescharacteristic
341C22#146811yescharacteristic
342C22#147311yescharacteristic
343C22#195511yescharacteristic
344C22#196411yescharacteristic
345C22#196711yescharacteristic
346C22C2×#35411yescharacteristic
347C22#212911yescharacteristic
348C22#213211yescharacteristic
349C22#213311yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—82no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—82no—
363C4—161no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2#965211yescharacteristic
369C2#1153111yescharacteristic
370C2#1153611yescharacteristic
371C2#1155111yescharacteristic
372C2#1155611yescharacteristic
373C2#1322411yescharacteristic
374C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#140032
C2×#13242
C2×#13192
#85582
#81532