Finite Group Databasebeta

← #100615

Subgroups — #100615

Order 512.

450 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 106 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#29868C211yesmaximal, characteristic
2#22652C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic, complemented
4#9898C211yesmaximal, characteristic, complemented
5#7574C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#476—21no—
9C2×#435—21no—
10C2×#424—21no—
11#2868C2211yescharacteristic
12#2293C2211yescharacteristic
13#2084—21no—
14#2084C412yes—
15#2071—21no—
16#2071C412yes—
17#1987—21no—
18#1940—21no—
19#1851—21no—
20#1850—21no—
21#1842C2211yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1798—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×M16—21no—
34C23×C8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4×D4D411yescharacteristic
39C2×#117—21no—
40#476—42no—
41#476—42no—
42#435—42no—
43#435—42no—
44C2×#106—22no—
45#424—42no—
46#424—42no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D411yescharacteristic
55C2×C4⋊C8D411yescharacteristic
56C2×C4⋊C8D411yescharacteristic
57C2×C4⋊C8D411yescharacteristic
58C2×#102—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×#97—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#384C2×C412yes—
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C43C2311yescharacteristic
76#340—22no—
77#340—22no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336C2×C412yes—
86C22×Q8—41no—
87C22×D4C22⋊C411yescharacteristic
88C23×C4—21no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—81no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—81no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C4⋊Q8—81no—
107C4⋊Q8—81no—
108C4⋊Q8—82no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4D812yes—
112C4×D4SD1612yes—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C42—21no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141#106—44no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8SD1612yes—
160#102—44no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#97—41no—
164#97—41no—
165C4×C8—81no—
166#95—21no—
167#95—21no—
168#95—21no—
169C24—21no—
170C2×Q8—82no—
171C2×Q8—82no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4#10212yes—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12311yescharacteristic
195M16—84no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
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—41no—
212C2×C8—84no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C4⋊C4—21no—
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—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4#10212yes—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—81no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42C2×D811yescharacteristic
254C42C2×SD1611yescharacteristic
255C42C2×SD1611yescharacteristic
256C42#13511yescharacteristic
257C42AGL(1,ℤ/8)11yescharacteristic
258C42AGL(1,ℤ/8)11yescharacteristic
259C42#13711yescharacteristic
260C42#13711yescharacteristic
261C23—22no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#38611yescharacteristic
269Q8—162no—
270Q8—162no—
271D4—42no—
272D4—81no—
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—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4C2×#10211yescharacteristic
303C2×C4#41711yescharacteristic
304C2×C4C4×SD1611yescharacteristic
305C2×C4#44011yescharacteristic
306C2×C4#44711yescharacteristic
307C2×C4#44811yescharacteristic
308C2×C4#45011yescharacteristic
309C2×C4#45211yescharacteristic
310C2×C4#45511yescharacteristic
311C2×C4#45611yescharacteristic
312C2×C4#46111yescharacteristic
313C2×C4#46311yescharacteristic
314C2×C4#47111yescharacteristic
315C2×C4#47211yescharacteristic
316C2×C4#48011yescharacteristic
317C2×C4#48111yescharacteristic
318C8—42no—
319C8—82no—
320C8—82no—
321C8—81no—
322C8—81no—
323C8—81no—
324C8—161no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#187411yescharacteristic
333C22#187711yescharacteristic
334C22#188211yescharacteristic
335C22#190311yescharacteristic
336C22#190711yescharacteristic
337C22#190911yescharacteristic
338C22#191511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—162no—
354C4#163911yescharacteristic
355C4#164311yescharacteristic
356C4#164711yescharacteristic
357C4#165111yescharacteristic
358C2—82no—
359C2#898511yescharacteristic
360C2#917311yescharacteristic
361C2#919511yescharacteristic
362C2#926911yescharacteristic
363C2#928711yescharacteristic
364C2#933811yescharacteristic
365C2#937511yescharacteristic
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. #9898C2semidirect
6. #7040C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#298682
#226522
#99052
#98982
#75742
#70402
#70222