Finite Group Databasebeta

← #103629

Subgroups — #103629

Order 512.

525 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 96 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#29755C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3C2×#2100C211yesmaximal, characteristic, complemented
4#9879C211yesmaximal, characteristic, complemented
5#8725C211yesmaximal, characteristic, complemented
6#7664C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic, complemented
8C2×#481—21no—
9#2856C2211yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#384C2211yescharacteristic
12#2100—22no—
13#2100—22no—
14#2084—21no—
15#2071—22no—
16#2065—22no—
17#2041—21no—
18#1957—21no—
19#1874—22no—
20#1874—21no—
21#1834—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24C4×C22⋊C8C2211yescharacteristic
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28#1296—22no—
29#1280—22no—
30#1270—22no—
31C23×C8—21no—
32C2×#127D411yescharacteristic
33C2×#121—21no—
34C2×#121—21no—
35C2×#121—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42C2311yescharacteristic
43#481—42no—
44#481—42no—
45C2×#106—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D412yes—
48C2×#102—22no—
49C2×#102—22no—
50C2×#102—24no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8D412yes—
54C2×C4×C8—21no—
55#390—21no—
56#390—21no—
57#390—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384D412yes—
66#384D411yescharacteristic
67C2×#95—21no—
68#340—22no—
69#340—24no—
70#340—24no—
71#340—24no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336D412yes—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—41no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—41no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95#127—42no—
96#121—41no—
97#121—41no—
98#121—42no—
99#121—41no—
100#121—41no—
101#121—42no—
102#121—82no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4SD1612yes—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C42—42no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135#106—44no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138#102—44no—
139#102—84no—
140#102—82no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C4×C8—42no—
147#95—41no—
148#95—41no—
149C24—41no—
150C24#12111yescharacteristic
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4—82no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4—82no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4#12011yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12711yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4#13511yescharacteristic
194C22×C4#13512yes—
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4#13711yescharacteristic
197C22×C4#13712yes—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C42—22no—
236C42—22no—
237C42—41no—
238C42—41no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23#39211yescharacteristic
251C23#46412yes—
252C23#46511yescharacteristic
253C23#46811yescharacteristic
254C23#47212yes—
255D4—84no—
256D4—82no—
257D4—82no—
258D4—162no—
259D4—162no—
260D4—162no—
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—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—164no—
294C2×C4—162no—
295C2×C4#44811yescharacteristic
296C2×C4#44911yescharacteristic
297C2×C4#45011yescharacteristic
298C2×C4#45212yes—
299C2×C4#45211yescharacteristic
300C2×C4#45412yes—
301C2×C4#45812yes—
302C2×C4#47112yes—
303C2×C4#49511yescharacteristic
304C2×C4#49711yescharacteristic
305C8—82no—
306C8—82no—
307C8—82no—
308C8—161no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22—161no—
321C22—161no—
322C22—161no—
323C22—161no—
324C22#167812yes—
325C22#169812yes—
326C22#200011yescharacteristic
327C22#200311yescharacteristic
328C22#200811yescharacteristic
329C22#201511yescharacteristic
330C22#201811yescharacteristic
331C22#201911yescharacteristic
332C22#202211yescharacteristic
333C4—22no—
334C4—21no—
335C4—21no—
336C4—22no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—82no—
341C4—161no—
342C4—161no—
343C4—162no—
344C4#161512yes—
345C4#161612yes—
346C2—22no—
347C2—22no—
348C2—321no—
349C2#992111yescharacteristic
350C2#995811yescharacteristic
351C2#999211yescharacteristic
352C2#1013011yescharacteristic
353C2#1015611yescharacteristic
354C2#1018011yescharacteristic
355C2#1026911yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2100C2semidirect
4. #9879C2semidirect
5. #8725C2semidirect
7. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#297552
#279732
C2×#21002
#98792
#87252
#76642
#70292