Finite Group Databasebeta

← #103713

Subgroups — #103713

Order 512.

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

normal subgroups only

This group has 551 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, complemented
2#28508C211yesmaximal, characteristic
3#28279C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9673C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic, complemented
8C2×#500—21no—
9C2×#474—22no—
10C2×#474—21no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×#127C2211yescharacteristic
14#2093—22no—
15#2083—22no—
16#2067—22no—
17#1990—21no—
18#1982—21no—
19#1957—21no—
20#1947—22no—
21#1947—21no—
22#1845—21no—
23#1842C2211yescharacteristic
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×Q8⋊C8—22no—
29#1296—22no—
30#1270—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×#127D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#500—42no—
41#500—42no—
42#474—44no—
43#474—42no—
44#474—42no—
45#474—44no—
46C2×#107—22no—
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×#102—22no—
56C2×#102—24no—
57C2×#102—22no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391—21no—
62#390—21no—
63#390—21no—
64#390—21no—
65#389—21no—
66#389—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384D411yescharacteristic
71#384D411yescharacteristic
72#384D412yes—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C43C2311yescharacteristic
76#340—22no—
77#340—24no—
78#336—21no—
79#336—21no—
80#336D412yes—
81Q8⋊C8—44no—
82C22×Q8—21no—
83C22×Q8—41no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C23×C4—41no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—41no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—42no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—42no—
100C4⋊Q8—41no—
101C4⋊Q8—82no—
102C4⋊Q8D812yes—
103#127—21no—
104#127—21no—
105#127—21no—
106#127Q1612yes—
107C4×Q8—42no—
108C4×Q8—42no—
109C4×D4—41no—
110C4×D4—42no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—82no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—41no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142#107—44no—
143#107—48no—
144#107—48no—
145C4⋊C8—24no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—42no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8D814yes—
155#102—44no—
156#102—48no—
157#102—82no—
158C4×C8—41no—
159C4×C8—42no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163#95—41no—
164#95—41no—
165C24—41no—
166C2×Q8—22no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—82no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—82no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—81no—
187C2×D4—81no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4#12011yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12011yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12711yescharacteristic
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—81no—
214C2×C8—81no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—44no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C22⋊C4—82no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—22no—
245C42—21no—
246C42—22no—
247C42—41no—
248C42—82no—
249C42C2×D811yescharacteristic
250C42C2×D812yes—
251C42C2×Q1611yescharacteristic
252C42#13511yescharacteristic
253C42#13511yescharacteristic
254C42AGL(1,ℤ/8)11yescharacteristic
255C42AGL(1,ℤ/8)11yescharacteristic
256C42AGL(1,ℤ/8)12yes—
257C42#13711yescharacteristic
258C42#13711yescharacteristic
259C23—42no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23#39211yescharacteristic
267Q8—44no—
268Q8—82no—
269Q8—162no—
270Q8—162no—
271D4—42no—
272D4—44no—
273D4—42no—
274D4—82no—
275D4—82no—
276D4—81no—
277D4—82no—
278D4—81no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—162no—
301C2×C4—162no—
302C2×C4#44711yescharacteristic
303C2×C4#44711yescharacteristic
304C2×C4#44712yes—
305C2×C4#44911yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#45111yescharacteristic
308C2×C4#45111yescharacteristic
309C2×C4#45211yescharacteristic
310C2×C4#45211yescharacteristic
311C2×C4#45511yescharacteristic
312C2×C4#45612yes—
313C2×C4#45611yescharacteristic
314C2×C4#45912yes—
315C2×C4#46611yescharacteristic
316C2×C4#46711yescharacteristic
317C2×C4#46911yescharacteristic
318C2×C4#47011yescharacteristic
319C2×C4#47212yes—
320C2×C4#49311yescharacteristic
321C2×C4#49611yescharacteristic
322C8—44no—
323C8—44no—
324C8—82no—
325C8—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22#199911yescharacteristic
334C22#200111yescharacteristic
335C22#200511yescharacteristic
336C22#200711yescharacteristic
337C22#200911yescharacteristic
338C22#201111yescharacteristic
339C22#201811yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—82no—
348C4—161no—
349C4—162no—
350C4—161no—
351C4#164111yescharacteristic
352C4#164411yescharacteristic
353C4#164712yes—
354C4#164912yes—
355C4#164911yescharacteristic
356C4#165211yescharacteristic
357C4#166712yes—
358C4#168712yes—
359C2—162no—
360C2#1007611yescharacteristic
361C2#1011811yescharacteristic
362C2#1019011yescharacteristic
363C2#1020411yescharacteristic
364C2#1023311yescharacteristic
365C2#1024311yescharacteristic
366C2#1026011yescharacteristic
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
1. #28689C2semidirect
4. #9898C2semidirect
5. #9673C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#286892
#285082
#282792
#98982
#96732
#75692
#70312