Finite Group Databasebeta

← #103694

Subgroups — #103694

Order 512.

515 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 114 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#28689C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#9865C211yesmaximal, characteristic
5#9692C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#500—21no—
9C2×#500—21no—
10C2×#462—21no—
11C2×#462—21no—
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2083—22no—
16#2083—22no—
17#2056—22no—
18#1990—21no—
19#1990—21no—
20#1947—22no—
21#1947—22no—
22#1842C2211yescharacteristic
23#1803—21no—
24#1803—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29#1279—22no—
30#1271—22no—
31C22×Q16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39#500—42no—
40#500—42no—
41#500—42no—
42#500—42no—
43#462—42no—
44#462—42no—
45#462—42no—
46#462—42no—
47C2×#107—24no—
48C2×#107—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D412yes—
54C2×C4⋊C8D412yes—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#391—21no—
61#391—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#384—21no—
67#384—21no—
68#384—21no—
69#384D411yescharacteristic
70#384D411yescharacteristic
71#384D412yes—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C43C2311yescharacteristic
77#340—22no—
78#340—24no—
79#340—24no—
80#336—21no—
81#336—21no—
82#336D412yes—
83#332—44no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—41no—
87C2×Q16—44no—
88C2×Q16—44no—
89C2×Q16—82no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109C4⋊Q8—41no—
110C4⋊Q8D812yes—
111C4⋊Q8D812yes—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147#107—48no—
148#107—48no—
149C4⋊C8—24no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—42no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8Q1614yescomplemented
159#103—44no—
160#103—44no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—42no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×Q8—81no—
185C2×Q8—81no—
186C2×Q8—81no—
187C2×Q8—81no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12711yescharacteristic
200Q16—88no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—81no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—81no—
234C4⋊C4—82no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—22no—
246C42—22no—
247C42—41no—
248C42—81no—
249C42—81no—
250C42—81no—
251C42—81no—
252C42C2×D811yescharacteristic
253C42C2×D811yescharacteristic
254C42C2×Q1612yes—
255C42#13511yescharacteristic
256C42#13511yescharacteristic
257C42AGL(1,ℤ/8)11yescharacteristic
258C42AGL(1,ℤ/8)11yescharacteristic
259C42AGL(1,ℤ/8)11yescharacteristic
260C42AGL(1,ℤ/8)11yescharacteristic
261C42#13712yes—
262C23#39211yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—42no—
266Q8—42no—
267Q8—81no—
268Q8—81no—
269Q8—81no—
270Q8—81no—
271Q8—161no—
272Q8—161no—
273Q8—161no—
274Q8—161no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4#44711yescharacteristic
300C2×C4#44711yescharacteristic
301C2×C4#44711yescharacteristic
302C2×C4#44711yescharacteristic
303C2×C4#44911yescharacteristic
304C2×C4#44911yescharacteristic
305C2×C4#44911yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#45112yes—
308C2×C4#45311yescharacteristic
309C2×C4#45411yescharacteristic
310C2×C4#45612yes—
311C2×C4#46212yes—
312C2×C4#46611yescharacteristic
313C2×C4#46611yescharacteristic
314C2×C4#46911yescharacteristic
315C2×C4#46911yescharacteristic
316C2×C4#47312yes—
317C2×C4#49411yescharacteristic
318C2×C4#49711yescharacteristic
319C8—44no—
320C8—44no—
321C8—82no—
322C8—82no—
323C22#199911yescharacteristic
324C22#199911yescharacteristic
325C22#200511yescharacteristic
326C22#200611yescharacteristic
327C22#200911yescharacteristic
328C22#201611yescharacteristic
329C22#202011yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—161no—
340C4—161no—
341C4—162no—
342C4#161911yescharacteristic
343C4#162411yescharacteristic
344C4#162511yescharacteristic
345C4#162511yescharacteristic
346C4#164812yes—
347C4#164912yes—
348C4#167412yes—
349C4#169412yes—
350C2#1006711yescharacteristic
351C2#1011711yescharacteristic
352C2#1018511yescharacteristic
353C2#1020311yescharacteristic
354C2#1022811yescharacteristic
355C2#1024311yescharacteristic
356C2#1026511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
158. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#286892
#282552
#98652
#96922
#75702
#70312