Finite Group Databasebeta

← #103719

Subgroups — #103719

Order 512.

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

normal subgroups only

This group has 611 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#28508C211yesmaximal, characteristic
2#28440C211yesmaximal, characteristic, complemented
3#28279C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#494—21no—
9C2×#474—22no—
10C2×#474—21no—
11C2×#462—22no—
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2093—22no—
16#2067—22no—
17#2016—22no—
18#1982—21no—
19#1957—21no—
20#1956—21no—
21#1947—21no—
22#1845—21no—
23#1842C2211yescharacteristic
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×D4⋊C8—22no—
29#1296—22no—
30#1271—22no—
31C22×Q16—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41#494—42no—
42#494—42no—
43#474—48no—
44#474—42no—
45#474—42no—
46#462—44no—
47#462—44no—
48C2×#107—22no—
49C2×#107—24no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D412yes—
55C2×C4⋊C8D412yes—
56C2×#103—24no—
57C2×#102—22no—
58C2×#102—24no—
59C2×#102—22no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#391—21no—
64#390—21no—
65#390—21no—
66#390—21no—
67#389—21no—
68#389—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72#384—21no—
73#384D411yescharacteristic
74#384D411yescharacteristic
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C43C2311yescharacteristic
78#340—22no—
79#336—21no—
80#336—21no—
81#336D412yes—
82D4⋊C8—44no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C23×C4—41no—
89C2×Q16—48no—
90C2×Q16—48no—
91C2×Q16—84no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—41no—
96C4⋊Q8—21no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—44no—
100C4⋊Q8—42no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—42no—
105C4⋊Q8—41no—
106C4⋊Q8D812yes—
107#127—21no—
108#127—21no—
109#127—21no—
110#127Q1612yescomplemented
111C4×Q8—42no—
112C4×Q8—42no—
113C4×D4—44no—
114C4×D4—42no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—42no—
119C4×D4—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C42—41no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146#107—44no—
147#107—48no—
148C4⋊C8—24no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—42no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8D814yes—
158#103—48no—
159#102—44no—
160#102—48no—
161#102—82no—
162C4×C8—41no—
163C4×C8—42no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167#95—41no—
168#95—41no—
169C24—41no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—44no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12711yescharacteristic
207Q16—816no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—81no—
219C2×C8—81no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4—82no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—82no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C22⋊C4—82no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C42—22no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—41no—
253C42—82no—
254C42C2×D811yescharacteristic
255C42C2×D812yes—
256C42C2×Q1611yescharacteristic
257C42#13511yescharacteristic
258C42#13511yescharacteristic
259C42AGL(1,ℤ/8)11yescharacteristic
260C42AGL(1,ℤ/8)11yescharacteristic
261C42AGL(1,ℤ/8)12yes—
262C42#13711yescharacteristic
263C42#13711yescharacteristic
264C23—42no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23#39211yescharacteristic
272Q8—44no—
273Q8—82no—
274Q8—84no—
275Q8—84no—
276D4—44no—
277D4—42no—
278D4—42no—
279D4—44no—
280D4—82no—
281D4—81no—
282D4—81no—
283D4—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—162no—
306C2×C4—162no—
307C2×C4#44711yescharacteristic
308C2×C4#44712yes—
309C2×C4#44711yescharacteristic
310C2×C4#44911yescharacteristic
311C2×C4#44911yescharacteristic
312C2×C4#45111yescharacteristic
313C2×C4#45111yescharacteristic
314C2×C4#45211yescharacteristic
315C2×C4#45211yescharacteristic
316C2×C4#45312yes—
317C2×C4#45511yescharacteristic
318C2×C4#45611yescharacteristic
319C2×C4#45912yes—
320C2×C4#46611yescharacteristic
321C2×C4#46711yescharacteristic
322C2×C4#46911yescharacteristic
323C2×C4#47011yescharacteristic
324C2×C4#47312yes—
325C2×C4#49311yescharacteristic
326C2×C4#49611yescharacteristic
327C8—44no—
328C8—44no—
329C8—82no—
330C8—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—161no—
335C22—161no—
336C22—161no—
337C22—161no—
338C22#199911yescharacteristic
339C22#200111yescharacteristic
340C22#200211yescharacteristic
341C22#200711yescharacteristic
342C22#200911yescharacteristic
343C22#201111yescharacteristic
344C22#202011yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—82no—
353C4—161no—
354C4—162no—
355C4—161no—
356C4#161912yes—
357C4#162112yes—
358C4#164111yescharacteristic
359C4#164411yescharacteristic
360C4#164911yescharacteristic
361C4#165211yescharacteristic
362C4#166912yes—
363C4#168912yes—
364C2—162no—
365C2#1004611yescharacteristic
366C2#1010711yescharacteristic
367C2#1018511yescharacteristic
368C2#1020411yescharacteristic
369C2#1023311yescharacteristic
370C2#1024811yescharacteristic
371C2#1027411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
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
2. #28440C2semidirect
4. #9898C2semidirect
6. #7570C2semidirect
7. #7031C2semidirect
110. #127Q16semidirect

Maximal subgroups

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

subgroupindex
#285082
#284402
#282792
#98982
#96742
#75702
#70312