Finite Group Databasebeta

← #190174

Subgroups — #190174

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#17365C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8757C218yesmaximal
5#8109C212yesmaximal
6#8105C212yesmaximal
7C2×#519C412yes—
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2351—21no—
12#2351—21no—
13#2351—22no—
14Q8×C4⋊C4—21no—
15Q8×C4⋊C4—21no—
16Q8×C4⋊C4—22no—
17#2316C412yes—
18#2316C2214yes—
19#2285C412yes—
20#2285C412yes—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1467—22no—
24#1467—22no—
25#1463C2214yes—
26#1463C2214yes—
27#1461—21no—
28#1461—21no—
29#1461—21no—
30#1461—21no—
31#1305C2218yes—
32#1305C2218yes—
33C23×Q8C2×C411yescharacteristic
34#519—48no—
35C2×C4×Q8—24no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8D411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C2×C4×Q8D412yes—
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D412yes—
48C2×#117C2×C412yes—
49C2×#117C2×C412yes—
50C2×#117C2×C414yes—
51C2×#117C2×C414yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C42C2×C412yes—
55C22×C42C2311yescharacteristic
56#432—22no—
57#432—22no—
58#432—22no—
59#432—22no—
60#431—22no—
61#431—22no—
62#431—22no—
63#431—22no—
64#423—22no—
65#423—22no—
66#407C2314yes—
67#407C2314yes—
68C4×M16—22no—
69C4×M16—22no—
70#389—24no—
71#389—22no—
72#389—22no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#384—22no—
78#384—22no—
79#384—21no—
80#384—21no—
81#382—22no—
82#382—22no—
83#382—21no—
84#382—21no—
85#380C2×C414yes—
86#380C2×C414yes—
87#380C2×C411yescharacteristic
88#380C2×C411yescharacteristic
89#380C2×C412yes—
90#380C2312yes—
91#380C2312yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100#324—24no—
101#324—24no—
102#324—24no—
103#324—24no—
104C22×Q8—24no—
105C22×Q8—24no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8C22⋊C412yes—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2411yescharacteristic
116C2×M16—24no—
117C2×M16—24no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—24no—
123C4×Q8—24no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—24no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—24no—
130C4×Q8—416no—
131C4×Q8—48no—
132C4×Q8—48no—
133#117—44no—
134#117—48no—
135#117—44no—
136#117—48no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4C22×C414yes—
160C2×C22⋊C4C22×C412yes—
161C2×C22⋊C4C22×C412yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C412yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C4⋊C8—42no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8—42no—
186C22⋊C8—24no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190#97—42no—
191#97—42no—
192C4×C8—42no—
193C4×C8—42no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—22no—
201C24C23×C411yescharacteristic
202C2×Q8—216no—
203C2×Q8—216no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8#10014yes—
212C2×Q8#10114yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—82no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4#11712yes—
238C22×C4#11712yes—
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4C2×#4112yes—
245M16—84no—
246M16—84no—
247C2×C8—44no—
248C2×C8—44no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—44no—
279C42—48no—
280C42—48no—
281C42—44no—
282C42#12312yes—
283C42#12312yes—
284C42#12312yes—
285C42#12312yes—
286C42#12312yes—
287C42#12312yes—
288C42#12312yes—
289C42#12312yes—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23C22×C22⋊C411yescharacteristic
295C23C2×#11711yescharacteristic
296C23C2×#11711yescharacteristic
297Q8—216no—
298Q8—28no—
299Q8—28no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4#38111yescharacteristic
336C2×C4#38111yescharacteristic
337C2×C4#38111yescharacteristic
338C2×C4#38112yes—
339C2×C4#38112yes—
340C2×C4#38111yescharacteristic
341C2×C4C2×#10014yes—
342C2×C4C2×#10012yes—
343C2×C4C2×#10114yes—
344C2×C4C2×#10112yes—
345C2×C4C2×#12312yes—
346C2×C4C2×#12312yes—
347C8—84no—
348C8—84no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—42no—
359C22C2×#38111yescharacteristic
360C22C22×#10011yescharacteristic
361C22C22×#10111yescharacteristic
362C4—22no—
363C4—24no—
364C4—24no—
365C4—22no—
366C4—22no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—84no—
373C4—82no—
374C4—82no—
375C4#182114yes—
376C4#183012yes—
377C2—21no—
378C2—21no—
379C2—41no—
380C2—41no—
381C2C2×#182111yescharacteristic
382C2#2071011yescharacteristic
383C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#341822
#173652
#87572
#81092
#81052