Finite Group Databasebeta

← #100614

Subgroups — #100614

Order 512.

435 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 98 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#29798C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic, complemented
4#9896C211yesmaximal, characteristic, complemented
5#7574C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#461—21no—
9C2×#433—21no—
10#2868C2211yescharacteristic
11#2293C2211yescharacteristic
12#2043—21no—
13#2043C412yes—
14#2013—21no—
15#2013C412yes—
16#1981—21no—
17#1955—21no—
18#1924—21no—
19#1923—21no—
20#1842C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1798—21no—
24#1771—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27#1296—21no—
28#1296—21no—
29#1296—21no—
30#1296—21no—
31#1271—21no—
32#1270—21no—
33C22×SD16—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8C2×C412yes—
37C2×C4×D4D411yescharacteristic
38C2×#117—21no—
39#461—42no—
40#461—42no—
41#433—42no—
42#433—42no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×#103—22no—
53C2×#103—22no—
54C2×#103—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×#97—21no—
60C2×#97—21no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#390C2×C411yescharacteristic
64#390D411yescharacteristic
65#389C2×C411yescharacteristic
66#389D411yescharacteristic
67#388—21no—
68#388—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72#384—21no—
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C43C2311yescharacteristic
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84C22×Q8—22no—
85C22×D4C22⋊C411yescharacteristic
86C23×C4—21no—
87C2×SD16—44no—
88C2×SD16—82no—
89C2×SD16—82no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—41no—
96C22×C8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—82no—
102C4⋊Q8—82no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107#117—41no—
108#117—41no—
109#117—41no—
110#117—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C42—21no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8SD1612yescomplemented
151#103—44no—
152#103—82no—
153#103—82no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—81no—
157C22⋊C8—81no—
158#97—42no—
159#97—41no—
160#97—41no—
161C4×C8—42no—
162C4×C8—81no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12311yescharacteristic
194SD16—88no—
195C2×C8—22no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42C2×SD1611yescharacteristic
251C42#13511yescharacteristic
252C42#13511yescharacteristic
253C42#13511yescharacteristic
254C42AGL(1,ℤ/8)11yescharacteristic
255C42AGL(1,ℤ/8)11yescharacteristic
256C42AGL(1,ℤ/8)11yescharacteristic
257C42#13711yescharacteristic
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23#38611yescharacteristic
266Q8—82no—
267Q8—82no—
268Q8—82no—
269Q8—82no—
270D4—42no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
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—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#41611yescharacteristic
302C2×C4#41811yescharacteristic
303C2×C4C4×SD1611yescharacteristic
304C2×C4#44011yescharacteristic
305C2×C4#44911yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#45011yescharacteristic
308C2×C4#45211yescharacteristic
309C2×C4#45511yescharacteristic
310C2×C4#45611yescharacteristic
311C2×C4#46111yescharacteristic
312C2×C4#46311yescharacteristic
313C2×C4#47111yescharacteristic
314C2×C4#47311yescharacteristic
315C2×C4#48211yescharacteristic
316C2×C4#48211yescharacteristic
317C8—42no—
318C8—82no—
319C8—82no—
320C8—81no—
321C8—81no—
322C8—81no—
323C8—161no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#187511yescharacteristic
332C22#187711yescharacteristic
333C22#188211yescharacteristic
334C22#190211yescharacteristic
335C22#190811yescharacteristic
336C22#191011yescharacteristic
337C22#191411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—162no—
353C4#164211yescharacteristic
354C4#164611yescharacteristic
355C4#165011yescharacteristic
356C4#165411yescharacteristic
357C2—161no—
358C2#898811yescharacteristic
359C2#917211yescharacteristic
360C2#919811yescharacteristic
361C2#926711yescharacteristic
362C2#928911yescharacteristic
363C2#933711yescharacteristic
364C2#937611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
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. #9905C2semidirect
4. #9896C2semidirect
6. #7040C2semidirect
7. #7022C2semidirect
150. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#297982
#227982
#99052
#98962
#75742
#70402
#70222