Finite Group Databasebeta

← #25013393

Subgroups — #25013393

Order 11008.

672 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 103 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#19137660C211yesmaximal, characteristic
2#19134994C211yesmaximal, characteristic
3#19134878C212yesmaximal, complemented
4#19132712C212yesmaximal, complemented
5C2×C43×#353C211yesmaximal, characteristic
6#14583374C2211yescharacteristic
7C2×D4×Dic43—22no—
8#14582904—21no—
9C2×#13063242C2211yescharacteristic
10#14582802—22no—
11#14582779—21no—
12#14582779C2212yes—
13#14582774—22no—
14C4×C4⋊Dic43—21no—
15#14582665—22no—
16#14582659—21no—
17C2×#127×C43C2211yescharacteristic
18C2×#99×C43C2212yes—
19C43×#353C414yescomplemented
20C23×Dic43—21no—
21C23×Dic43D412yes—
22C2×#10621877—21no—
23D4×Dic43—41no—
24D4×Dic43—41no—
25D4×Dic43—48no—
26#13063284—21no—
27C2×C4⋊Dic43—21no—
28C2×C4⋊Dic43—21no—
29C2×C4⋊Dic43—42no—
30C2×C22⋊Dic43—22no—
31C2×C22⋊Dic43—22no—
32C2×C22⋊Dic43—22no—
33C2×C22⋊Dic43—21no—
34C2×C22⋊Dic43D412yes—
35C2×C4×Dic43—22no—
36#13063246—41no—
37#13063246—41no—
38#13063242—22no—
39C43⋊#95—22no—
40C43⋊#95—42no—
41C22×D4×C43—22no—
42C22×D4×C43D411yescharacteristic
43C22×D4×C43C2311yescharacteristic
44#127×C43—22no—
45#127×C43C2×C412yes—
46#127×C43Q812yes—
47C2×C22⋊C4×C43—22no—
48C2×C42×C43D411yescharacteristic
49#99×C43—22no—
50#99×C43C2×C414yescomplemented
51C24×C43—21no—
52C24×C43C2×D412yes—
53C2×D4×C43—21no—
54C2×D4×C43—24no—
55C2×D4×C43—24no—
56C2×D4×C43—41no—
57C2×D4×C43—41no—
58C2×D4×C43—44no—
59C2×D4×C43—44no—
60C2×D4×C43—41no—
61C2×D4×C43—41no—
62C2×D4×C43C4⋊C412yescomplemented
63C2×D4×C43C22×C411yescharacteristic
64C2×D4×C43C2×Q811yescharacteristic
65C2×D4×C43#4112yes—
66C22×C4×C43—21no—
67C22×C4×C43—42no—
68C22×C4×C43C2×D411yescharacteristic
69C22×Dic43—22no—
70C22×Dic43—22no—
71C22×Dic43—22no—
72C22×Dic43—22no—
73C22×Dic43—42no—
74C22×Dic43—44no—
75C22×Dic43—44no—
76C22×Dic43—81no—
77C22⋊C4×C43—24no—
78C22⋊C4×C43—42no—
79C42×C43—21no—
80C42×C43C4⋊C412yescomplemented
81C22⋊Dic43—42no—
82C22⋊Dic43—42no—
83C22⋊Dic43—44no—
84C22⋊Dic43—44no—
85C22⋊Dic43—41no—
86C22⋊Dic43—81no—
87C4⋊Dic43—24no—
88C4⋊Dic43—21no—
89C4⋊Dic43—21no—
90C4⋊Dic43—42no—
91C4⋊Dic43—84no—
92C4×Dic43—44no—
93C4×Dic43—41no—
94#10621877—41no—
95#10621877—41no—
96C2×C43⋊C8—41no—
97C23×C43—22no—
98C23×C43—22no—
99C23×C43—22no—
100C23×C43—42no—
101C23×C43—42no—
102C23×C43—42no—
103C23×C43—42no—
104C23×C43—41no—
105C23×C43—41no—
106C23×C43C4×D412yes—
107C23×C43#12011yescharacteristic
108C23×C43C22⋊Q812yes—
109D4×C43—41no—
110D4×C43—41no—
111D4×C43—42no—
112D4×C43—42no—
113D4×C43—48no—
114D4×C43—48no—
115D4×C43—81no—
116C2×C4×C43—22no—
117C2×C4×C43—22no—
118C2×C4×C43—22no—
119C2×C4×C43—21no—
120C2×C4×C43—21no—
121C2×C4×C43—44no—
122C2×C4×C43—82no—
123C2×C4×C43—82no—
124C2×C4×C43C2×C4⋊C411yescharacteristic
125C2×C4×C43#12311yescharacteristic
126C2×Dic43—24no—
127C2×Dic43—42no—
128C2×Dic43—42no—
129C2×Dic43—44no—
130C2×Dic43—42no—
131C2×Dic43—42no—
132C2×Dic43—41no—
133C2×Dic43—41no—
134C2×Dic43—42no—
135C2×Dic43—82no—
136C2×Dic43—84no—
137C2×Dic43—82no—
138C2×Dic43—82no—
139C43⋊C8—81no—
140#12964—431nomaximal, Sylow
141C22×C43—21no—
142C22×C43—21no—
143C22×C43—22no—
144C22×C43—22no—
145C22×C43—42no—
146C22×C43—42no—
147C22×C43—44no—
148C22×C43—42no—
149C22×C43—42no—
150C22×C43—42no—
151C22×C43—82no—
152C22×C43—82no—
153C22×C43—81no—
154C22×C43—81no—
155C22×C43#38511yescharacteristic
156C22×C43B(4,𝔽2)11yescharacteristic
157C22×C43#45811yescharacteristic
158C4×C43—24no—
159C4×C43—21no—
160C4×C43—21no—
161C4×C43—84no—
162Dic43—44no—
163Dic43—82no—
164Dic43—81no—
165Dic43—81no—
166#2366—431no—
167C2×#353D4311yescharacteristic
168#1887—431no—
169#1867—432no—
170#1390—432no—
171C2×C43—21no—
172C2×C43—21no—
173C2×C43—42no—
174C2×C43—42no—
175C2×C43—82no—
176C2×C43#186711yescharacteristic
177C2×C43#219611yescharacteristic
178C2×C43#219811yescharacteristic
179C2×#127C2×D4311yescharacteristic
180#518—431no—
181C2×C4×D4—862no—
182#418—861no—
183C2×#100—431no—
184C2×#99C2×D4312yes—
185#390—862no—
186#386—432no—
187#386—861no—
188#385—862no—
189C4×C4⋊C4—861no—
190#353Dic4314yescomplemented
191#342—862no—
192#339—861no—
193C43#1296411yescharacteristic, complemented, Sylow
194C22×D4—22no—
195C22×D4D17211yescharacteristic
196C22×D4C22×D4311yescharacteristic
197C23×C4—432no—
198C23×C4—861no—
199C2×M16—861no—
200#127—22no—
201#127Dic8612yes—
202#127C2×Dic4312yes—
203C4×D4—1721no—
204C4×D4—1721no—
205C4×D4—1728no—
206#117—861no—
207C2×C4⋊C4—861no—
208C2×C4⋊C4—861no—
209C2×C4⋊C4—1722no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—432no—
212C2×C22⋊C4—861no—
213C2×C22⋊C4—862no—
214C2×C22⋊C4—862no—
215C2×C22⋊C4—862no—
216C2×C42—862no—
217C2×C42D17211yescharacteristic
218#102—1721no—
219#102—1721no—
220#100—862no—
221#99—22no—
222#99C2×Dic4314yescomplemented
223#95—862no—
224#95—1722no—
225C24—21no—
226C24D4×D4312yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4C4⋊Dic4312yescomplemented
237C2×D4C2×Dic8611yescharacteristic
238C2×D4#1062190012yes—
239C2×D4C22×Dic4311yescharacteristic
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—862no—
243C22×C4—862no—
244C22×C4—862no—
245C22×C4—862no—
246C22×C4—3441no—
247C22×C4—1724no—
248C22×C4—1724no—
249C22×C4—1722no—
250C22×C4C2×D17211yescharacteristic
251M16—1721no—
252M16—1721no—
253C2×C8—1721no—
254C4⋊C4—864no—
255C4⋊C4—861no—
256C4⋊C4—861no—
257C4⋊C4—3444no—
258C4⋊C4—1722no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—3441no—
262C22⋊C4—1724no—
263C22⋊C4—1724no—
264C22⋊C4—1722no—
265C22⋊C4—1721no—
266C22⋊C4—1722no—
267C42—21no—
268C42—1724no—
269C42—1721no—
270C42C4⋊Dic4312yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23D4×Dic4312yes—
281C23#1306329511yescharacteristic
282C23#1306330212yes—
283D4—41no—
284D4—42no—
285D4—41no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—864no—
299C2×C4—3442no—
300C2×C4—3442no—
301C2×C4—3442no—
302C2×C4—3444no—
303C2×C4—1722no—
304C2×C4—1722no—
305C2×C4—1721no—
306C2×C4—1721no—
307C2×C4—1722no—
308C2×C4—1722no—
309C2×C4—1722no—
310C2×C4—1724no—
311C2×C4C2×C4⋊Dic4311yescharacteristic
312C2×C4#1306330911yescharacteristic
313C8—3441no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1458277411yescharacteristic
329C22#1458308711yescharacteristic
330C22#1458309011yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—84no—
335C4—3442no—
336C4—3441no—
337C4—3441no—
338C4—1724no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#1913487811yescharacteristic
345C2#1913658111yescharacteristic
346C2#1913659311yescharacteristic
subgroupG/H
centerC22#14583087
commutatorC2×D4×C43C22×C4
FrattiniC22×D4C22×D43
FittingC2×C43×#353C2
socleC22×C43B(4,𝔽2)
prime radicalC2×#127×C43C22
cosocleC22×D4×C43C23
hypercenterC2×#353D43
hypocenterC43#12964
LCSG ▷ C2×D4×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #19134878C2semidirect
4. #19132712C2semidirect
19. C43×#353C4semidirect
50. #99×C43C2×C4semidirect
62. C2×D4×C43C4⋊C4semidirect
80. C42×C43C4⋊C4semidirect
190. #353Dic43semidirect
193. C43#12964semidirect
222. #99C2×Dic43semidirect
236. C2×D4C4⋊Dic43semidirect
270. C42C4⋊Dic43semidirect

Maximal subgroups

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

subgroupindex
#191376602
#191349942
#191348782
#191327122
C2×C43×#3532
#1296443