Finite Group Databasebeta

← #101447

Subgroups — #101447

Order 512.

487 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 104 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#29254C211yesmaximal, characteristic
2#22260C211yesmaximal, characteristic
3#9643C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic, complemented
6#7040C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×C8×D4—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2093C412yes—
12#2023C412yes—
13#1982—21no—
14#1954—21no—
15#1947—21no—
16#1869—22no—
17#1869—22no—
18#1862—22no—
19#1842C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1744C2211yescharacteristic
24C2×#331—21no—
25C2×Q8⋊C8—21no—
26C2×Q8⋊C8—21no—
27#1296—21no—
28#1271—22no—
29C23×C8—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#127D411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4D412yes—
36C8×D4—42no—
37C8×D4—42no—
38C2×#107—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#102—22no—
46C2×#102—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#391—21no—
53#390—21no—
54#390—21no—
55#390C2×C412yes—
56#389—21no—
57#389—21no—
58#384C2×C411yescharacteristic
59#384D411yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C43C2311yescharacteristic
63#340—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336C2×C412yes—
71#336D412yes—
72#331—42no—
73Q8⋊C8—42no—
74Q8⋊C8—42no—
75C22×Q8—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4C22⋊C412yes—
79C23×C4—22no—
80C22×C8—21no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—44no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—82no—
88C22×C8—82no—
89C4⋊Q8—42no—
90#127—21no—
91#127—21no—
92#127SD1612yes—
93#127Q1612yes—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—82no—
103C4×D4—82no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4C22⋊C412yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C42—41no—
117C2×C42C22×C411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124#107—82no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—81no—
136#102—44no—
137#102—82no—
138C22⋊C8—44no—
139C22⋊C8—82no—
140C22⋊C8—82no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8D812yes—
145#95—41no—
146#95—41no—
147C24—22no—
148C2×Q8—22no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4#10314yes—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—41no—
196C2×C8—88no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—21no—
223C42—21no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—81no—
229C42—81no—
230C42C2×D811yescharacteristic
231C42C2×SD1611yescharacteristic
232C42C2×Q1611yescharacteristic
233C42#13511yescharacteristic
234C42#13511yescharacteristic
235C42#13511yescharacteristic
236C42AGL(1,ℤ/8)11yescharacteristic
237C42AGL(1,ℤ/8)11yescharacteristic
238C23—24no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23#38611yescharacteristic
246Q8—42no—
247Q8—42no—
248Q8—81no—
249Q8—81no—
250D4—44no—
251D4—42no—
252D4—42no—
253D4—42no—
254D4—42no—
255D4—82no—
256D4—82no—
257D4—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4C2×#10311yescharacteristic
286C2×C4#41611yescharacteristic
287C2×C4C4×D811yescharacteristic
288C2×C4#44211yescharacteristic
289C2×C4#44711yescharacteristic
290C2×C4#44911yescharacteristic
291C2×C4#45911yescharacteristic
292C2×C4#46311yescharacteristic
293C2×C4#46511yescharacteristic
294C2×C4#46511yescharacteristic
295C2×C4#46711yescharacteristic
296C2×C4#46711yescharacteristic
297C2×C4#47111yescharacteristic
298C2×C4#47311yescharacteristic
299C8—42no—
300C8—82no—
301C8—81no—
302C8—82no—
303C8—161no—
304C8—161no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22#187911yescharacteristic
313C22#188111yescharacteristic
314C22#189111yescharacteristic
315C22#189211yescharacteristic
316C22#190411yescharacteristic
317C22#190811yescharacteristic
318C22#191111yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—161no—
332C4#166511yescharacteristic
333C4#166911yescharacteristic
334C4#167911yescharacteristic
335C4#168311yescharacteristic
336C2—84no—
337C2#904911yescharacteristic
338C2#921911yescharacteristic
339C2#929911yescharacteristic
340C2#930311yescharacteristic
341C2#937311yescharacteristic
342C2#948311yescharacteristic
343C2#952711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9643C2semidirect
5. #7652C2semidirect
6. #7040C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#292542
#222602
#96432
#85882
#76522
#70402
#70182