Finite Group Databasebeta

← #25012586

Subgroups — #25012586

Order 11008.

588 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 101 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#19137321C211yesmaximal, characteristic
2C2×#14582692C211yesmaximal, characteristic
3#19134581C212yesmaximal, complemented
4#19132719C212yesmaximal, complemented
5C2×C43×#353C211yesmaximal, characteristic
6C2×D4×Dic43C411yescharacteristic, complemented
7C2×D4×Dic43C2211yescharacteristic
8C2×#13063240C2212yes—
9#14582802C411yescharacteristic, complemented
10#14582802C2211yescharacteristic
11#14582779—22no—
12#14582779—22no—
13#14582765C411yescharacteristic, complemented
14#14582757—22no—
15#14582745C411yescharacteristic, complemented
16#14582692—22no—
17#14582665—22no—
18C2×#127×C43C2211yescharacteristic
19C2×#99×C43C2212yes—
20C43×#353—22no—
21C23×Dic43—21no—
22C23×Dic43D412yes—
23D4×Dic43—42no—
24D4×Dic43—42no—
25C2×C4⋊Dic43C2×C411yescharacteristic
26C2×C4⋊Dic43C2×C411yescharacteristic
27C2×Dic43⋊C4—42no—
28C2×C22⋊Dic43—21no—
29C2×C22⋊Dic43—22no—
30C2×C22⋊Dic43—22no—
31C2×C22⋊Dic43—22no—
32C2×C22⋊Dic43D412yes—
33C2×C4×Dic43C2×C411yescharacteristic
34C2×C4×Dic43C2×C411yescharacteristic
35#13063240—22no—
36#13063240—22no—
37C43⋊#95—21no—
38C43⋊#95—21no—
39C43⋊#95—22no—
40C43⋊#95—42no—
41C22×D4×C43—22no—
42C22×D4×C43C2×C411yescharacteristic
43C22×D4×C43C2311yescharacteristic
44#127×C43—22no—
45#127×C43D412yes—
46#127×C43Q812yes—
47C2×C22⋊C4×C43—22no—
48C2×C42×C43C2×C411yescharacteristic
49#99×C43—22no—
50#99×C43—22no—
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—41no—
59C2×D4×C43—41no—
60C2×D4×C43—82no—
61C2×D4×C43—82no—
62C2×D4×C43C4⋊C412yescomplemented
63C2×D4×C43C2×D411yescharacteristic
64C2×D4×C43C2×Q811yescharacteristic
65C2×D4×C43#4112yes—
66C22×C4×C43—21no—
67C22×C4×C43—42no—
68C22×C4×C43C22×C411yescharacteristic
69C22×Dic43—22no—
70C22×Dic43—22no—
71C22×Dic43—21no—
72C22×Dic43—21no—
73C22×Dic43—42no—
74C22×Dic43—42no—
75C22×Dic43—82no—
76C22×Dic43—82no—
77C22×Dic43C22⋊C411yescharacteristic
78C22×Dic43C22⋊C412yes—
79C22×Dic43C22⋊C411yescharacteristic
80C22⋊C4×C43—42no—
81C22⋊C4×C43—42no—
82C42×C43—21no—
83C42×C43C4⋊C412yescomplemented
84C22⋊Dic43—42no—
85C22⋊Dic43—42no—
86C22⋊Dic43—42no—
87C22⋊Dic43—42no—
88C22⋊Dic43—82no—
89C22⋊Dic43—82no—
90C4⋊Dic43—41no—
91C4⋊Dic43—41no—
92Dic43⋊C4—84no—
93C4×Dic43—22no—
94C4×Dic43—41no—
95C23×C43—22no—
96C23×C43—22no—
97C23×C43—22no—
98C23×C43—42no—
99C23×C43—42no—
100C23×C43—42no—
101C23×C43—42no—
102C23×C43—41no—
103C23×C43—41no—
104C23×C43C2×C22⋊C411yescharacteristic
105C23×C43#12112yes—
106C23×C43C22⋊Q812yes—
107D4×C43—41no—
108D4×C43—41no—
109D4×C43—42no—
110D4×C43—42no—
111D4×C43—42no—
112D4×C43—84no—
113D4×C43—84no—
114C2×C4×C43—21no—
115C2×C4×C43—21no—
116C2×C4×C43—22no—
117C2×C4×C43—41no—
118C2×C4×C43—41no—
119C2×C4×C43—82no—
120C2×C4×C43—82no—
121C2×C4×C43—82no—
122C2×C4×C43C2×C4⋊C411yescharacteristic
123C2×C4×C43#11711yescharacteristic
124C2×Dic43—21no—
125C2×Dic43—22no—
126C2×Dic43—41no—
127C2×Dic43—44no—
128C2×Dic43—42no—
129C2×Dic43—41no—
130C2×Dic43—41no—
131C2×Dic43—41no—
132C2×Dic43—42no—
133C2×Dic43—41no—
134C2×Dic43—41no—
135C2×Dic43—82no—
136C2×Dic43—84no—
137C2×Dic43—82no—
138C2×Dic43—82no—
139C2×Dic43—82no—
140C2×Dic43#9912yes—
141#12799—431nomaximal, Sylow
142C22×C43—21no—
143C22×C43—21no—
144C22×C43—22no—
145C22×C43—22no—
146C22×C43—42no—
147C22×C43—42no—
148C22×C43—44no—
149C22×C43—42no—
150C22×C43—42no—
151C22×C43—42no—
152C22×C43—82no—
153C22×C43—82no—
154C22×C43—81no—
155C22×C43—81no—
156C22×C43#38011yescharacteristic
157C22×C43C2×#9911yescharacteristic
158C22×C43#41011yescharacteristic
159C4×C43—21no—
160C4×C43—21no—
161C4×C43—42no—
162C4×C43—162no—
163Dic43—22no—
164Dic43—42no—
165Dic43—41no—
166Dic43—81no—
167Dic43—81no—
168Dic43—162no—
169#2328—431no—
170C2×#353—431no—
171C2×#353D4311yescharacteristic
172#1799—432no—
173#1393—432no—
174C2×C43—21no—
175C2×C43—21no—
176C2×C43—42no—
177C2×C43—42no—
178C2×C43—82no—
179C2×C43#179911yescharacteristic
180C2×C43#212711yescharacteristic
181C2×C43#212811yescharacteristic
182C2×#127C2×D4311yescharacteristic
183C2×C4×D4—431no—
184C2×C4×D4—431no—
185C2×#99—432no—
186C2×#99C2×D4312yes—
187#390—431no—
188#390—431no—
189#386—862no—
190#386—862no—
191#382—431no—
192#380—862no—
193#376—431no—
194#353—22no—
195#353—862no—
196#342—862no—
197C43#1279911yescharacteristic, complemented, Sylow
198C22×D4—22no—
199C22×D4C4×D4311yescharacteristic
200C22×D4C22×D4311yescharacteristic
201C23×C4—432no—
202C23×C4—861no—
203#127—22no—
204#127Dic8612yes—
205#127C22⋊D4312yes—
206C4×D4—1722no—
207C4×D4—1722no—
208C2×C4⋊C4—431no—
209C2×C4⋊C4—431no—
210C2×C4⋊C4—1722no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—432no—
213C2×C22⋊C4—862no—
214C2×C22⋊C4—862no—
215C2×C22⋊C4—861no—
216C2×C22⋊C4—862no—
217C2×C42—431no—
218C2×C42—431no—
219C2×C42C4×D4311yescharacteristic
220#99—22no—
221#99—22no—
222#99—862no—
223#99—862no—
224#95—861no—
225#95—861no—
226#95—862no—
227#95—1722no—
228C24—21no—
229C24D4×D4312yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4Dic43⋊C412yescomplemented
240C2×D4C2×Dic8611yescharacteristic
241C2×D4#1062190012yes—
242C2×D4C2×C22⋊D4311yescharacteristic
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—431no—
246C22×C4—431no—
247C22×C4—432no—
248C22×C4—862no—
249C22×C4—862no—
250C22×C4—861no—
251C22×C4—861no—
252C22×C4—3442no—
253C22×C4—3442no—
254C22×C4—1722no—
255C22×C4—1722no—
256C22×C4C2×C4×D4311yescharacteristic
257C4⋊C4—3444no—
258C4⋊C4—1721no—
259C4⋊C4—1721no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—3442no—
263C22⋊C4—3442no—
264C22⋊C4—1722no—
265C22⋊C4—1722no—
266C22⋊C4—1722no—
267C22⋊C4—1722no—
268C42—21no—
269C42—862no—
270C42—1721no—
271C42Dic43⋊C412yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23C22⋊C4×D4311yescharacteristic
282C23#1306329712yes—
283C23#1306330212yes—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—432no—
300C2×C4—861no—
301C2×C4—862no—
302C2×C4—3442no—
303C2×C4—3444no—
304C2×C4—3442no—
305C2×C4—3442no—
306C2×C4—3442no—
307C2×C4—1721no—
308C2×C4—1724no—
309C2×C4—1721no—
310C2×C4—1721no—
311C2×C4—1722no—
312C2×C4—1721no—
313C2×C4—1721no—
314C2×C4—1722no—
315C2×C4—1721no—
316C2×C4C2×Dic43⋊C411yescharacteristic
317C2×C4#1306328111yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22#1458275711yescharacteristic
333C22#99×D4311yescharacteristic
334C22#1458287111yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—162no—
339C4—862no—
340C4—3441no—
341C4—3441no—
342C4—1722no—
343C4—1721no—
344C4—6882no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#1913458111yescharacteristic
351C2#1913626111yescharacteristic
352C2#1913626511yescharacteristic
subgroupG/H
centerC22#99×D43
commutatorC22×C4×C43C22×C4
FrattiniC22×D4C22×D43
FittingC2×C43×#353C2
socleC22×C43C2×#99
prime radicalC2×#127×C43C22
cosocleC22×D4×C43C23
hypercenterC2×#353D43
hypocenterC43#12799
LCSG ▷ C22×C4×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. #19134581C2semidirect
4. #19132719C2semidirect
6. C2×D4×Dic43C4semidirect
9. #14582802C4semidirect
13. #14582765C4semidirect
15. #14582745C4semidirect
62. C2×D4×C43C4⋊C4semidirect
83. C42×C43C4⋊C4semidirect
197. C43#12799semidirect
239. C2×D4Dic43⋊C4semidirect
271. C42Dic43⋊C4semidirect

Maximal subgroups

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

subgroupindex
#191373212
C2×#145826922
#191345812
#191327192
C2×C43×#3532
#1279943