Finite Group Databasebeta

← #25065174

Subgroups — #25065174

Order 11008.

950 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D43C211yesmaximal, characteristic
2#19143227C212yesmaximal
3#19133333C211yesmaximal, characteristic, complemented
4#19133332C212yesmaximal, complemented
5#19133310C214yesmaximal, complemented
6#19133258C214yesmaximal, complemented
7C43×#1561C211yesmaximal, characteristic, complemented
8#14583456C2211yescharacteristic
9#14583455C412yes—
10#14583454C2212yes—
11#14583453—22no—
12#14583452C412yes—
13#14583451—22no—
14C4⋊Q8×D43C2212yes—
15#125×D43C412yes—
16C22⋊Q8×D43—22no—
17C4×Q8×D43—22no—
18#117×D43C412yes—
19C2×C42×D43C2211yescharacteristic
20#14582955—22no—
21#14582955C2214yes—
22#14582951—22no—
23#14582951C2214yes—
24#14582923C2212yes—
25#14582862—22no—
26M16×Dic43—22no—
27#14582646C2214yescomplemented
28#14582645C2214yescomplemented
29#14582644C2214yescomplemented
30C43×#533C2211yescharacteristic
31C43×#423C2212yes—
32C43×#332C2214yescomplemented
33C2×Q8×D43—22no—
34C22×C4×D43C2×C411yescharacteristic
35C22×C4×D43D412yes—
36C2×#10621877—22no—
37#13063325C2312yes—
38#13063324—22no—
39#13063324C2×C414yes—
40#13063323—22no—
41#13063317C2×C412yes—
42#13063316C2×C414yes—
43#13063315—24no—
44#13063305—22no—
45#13063304—22no—
46#13063303—24no—
47#13063302—24no—
48#13063301—22no—
49Q8×Dic43—22no—
50C4×Dic86—22no—
51C4×Dic86—24no—
52#13063290—24no—
53#13063290—22no—
54#13063284C2×C412yes—
55#13063283C2×C414yes—
56#13063282—22no—
57#13063281—22no—
58C4⋊C4×D43—24no—
59C4⋊C4×D43—22no—
60C4⋊C4×D43C2×C414yes—
61C22⋊C4×D43—22no—
62C2×C4×Dic43C2×C411yescharacteristic
63C2×C4×Dic43D412yes—
64C42×D43—22no—
65C42×D43C2×C411yescharacteristic
66C42×D43C2×C411yescharacteristic
67C42×D43D412yes—
68C42×D43D412yes—
69C42×D43C2311yescharacteristic
70C42×D43C2311yescharacteristic
71C4⋊(C43⋊C8)—22no—
72C4⋊(C43⋊C8)C2314yes—
73(C2×D43)⋊C8—24no—
74(C2×D43)⋊C8—24no—
75#13063235—24no—
76C8×Dic43—24no—
77C2×M16×C43—22no—
78C4⋊Q8×C43C2312yes—
79#125×C43C2×C412yes—
80C22⋊Q8×C43—22no—
81C4×Q8×C43—22no—
82#117×C43C2×C412yes—
83C2×C42×C43C2311yescharacteristic
84C4⋊C8×C43—22no—
85C4⋊C8×C43C2314yes—
86C23×D43C22⋊C411yescharacteristic
87C2×Q8×C43—22no—
88C22×C4×C43C22×C411yescharacteristic
89C22×C4×C43C2×D412yes—
90C22×Dic43C22⋊C411yescharacteristic
91C22×Dic43C22⋊C412yes—
92Q8×D43—44no—
93Q8×D43—44no—
94C2×C4×D43—24no—
95C2×C4×D43—24no—
96C2×C4×D43—42no—
97C2×C4×D43—42no—
98C2×C4×D43C22⋊C411yescharacteristic
99C2×C4×D43C22⋊C412yes—
100C2×C4×D43C22⋊C412yes—
101C2×C4×D43C22⋊C411yescharacteristic
102C2×C4×D43C22×C411yescharacteristic
103C2×C4×D43C22×C411yescharacteristic
104C2×C4×D43C2×D412yes—
105C2×C4×D43C2×D412yes—
106C2×Dic86—22no—
107C2×Dic86—24no—
108M16×C43—44no—
109C2×C8×C43—24no—
110C4⋊C4×C43—24no—
111C4⋊C4×C43—22no—
112C4⋊C4×C43C22×C414yes—
113C22⋊C4×C43—22no—
114C42×C43—22no—
115C42×C43C22×C411yescharacteristic
116C42×C43C2×D412yes—
117C42×C43C2411yescharacteristic
118C22⋊Dic43—22no—
119(C2×D43)⋊C4—24no—
120C4⋊Dic43—22no—
121C4⋊Dic43—24no—
122C4⋊Dic43C22×C414yes—
123Dic43⋊C4—24no—
124Dic43⋊C4—24no—
125Dic43⋊C4—24no—
126Dic43⋊C4—24no—
127C4×Dic43—22no—
128C4×Dic43—24no—
129C4×Dic43C22⋊C412yes—
130C4×Dic43C22⋊C412yes—
131C4×Dic43C22⋊C412yes—
132C4×Dic43C22×C411yescharacteristic
133C4×Dic43C22×C411yescharacteristic
134C4×Dic43C2×D412yes—
135C4×Dic43C2×D412yes—
136#10621877—44no—
137C2×C43⋊C8—24no—
138C23×C43C2×C22⋊C411yescharacteristic
139C22×D43—22no—
140C22×D43—42no—
141C22×D43C2×C22⋊C411yescharacteristic
142C22×D43C2×C22⋊C411yescharacteristic
143Q8×C43—42no—
144Q8×C43—42no—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—24no—
149C2×C4×C43C2×C22⋊C412yes—
150C2×C4×C43C2×C22⋊C411yescharacteristic
151C2×C4×C43C23×C411yescharacteristic
152C2×C4×C43C22×D412yes—
153C2×Dic43—22no—
154C2×Dic43—24no—
155C2×Dic43—42no—
156C2×Dic43C2×C22⋊C412yes—
157C2×Dic43C2×C22⋊C412yes—
158C2×Dic43C2×C22⋊C411yescharacteristic
159C2×Dic43C2×C22⋊C411yescharacteristic
160C4×D43—22no—
161C4×D43—22no—
162C4×D43—22no—
163C4×D43—22no—
164C4×D43—22no—
165C4×D43—22no—
166C4×D43—22no—
167C4×D43—42no—
168C4×D43—42no—
169C4×D43—48no—
170C4×D43#10114yes—
171Dic86—48no—
172Dic86—42no—
173Dic86—42no—
174C8×C43—44no—
175C43⋊C8—44no—
176#17739—431nomaximal, Sylow
177C22×C43—21no—
178C22×C43—22no—
179C22×C43C22×C22⋊C411yescharacteristic
180C2×D43—22no—
181C2×D43—21no—
182C2×D43—22no—
183C2×D43—44no—
184C2×D43C2×#10112yes—
185C4×C43—21no—
186C4×C43—22no—
187C4×C43—44no—
188C4×C43C2×#10112yes—
189C4×C43#57512yes—
190C4×C43#57512yes—
191Dic43—21no—
192Dic43—22no—
193Dic43—22no—
194Dic43—44no—
195Dic43C2×#10112yes—
196C2×#533—431no—
197#2920—432no—
198#1561—432no—
199#1561—431no—
200#1561D4311yescharacteristic, complemented
201#1556—434no—
202#1544—434no—
203C2×C43—41no—
204C2×C43C22×#10111yescharacteristic
205C2×C43#289612yes—
206D43—22no—
207D43—41no—
208#533—432no—
209#533—432no—
210#533—432no—
211#533—431no—
212#533—862no—
213#533—862no—
214#533C2×D4311yescharacteristic
215C2×C4⋊Q8—432no—
216C2×#125—432no—
217C2×C22⋊Q8—862no—
218C2×C4×Q8—862no—
219C2×#117—432no—
220C22×C42—431no—
221#432—434no—
222#432—862no—
223#431—434no—
224#431—862no—
225#423—432no—
226#423C2×D4312yes—
227#407—862no—
228C4×M16—862no—
229#332—434no—
230#332—434no—
231#332—434no—
232#332C2×D4314yescomplemented
233C43#1773911yescharacteristic, complemented, Sylow
234C22×Q8—862no—
235C23×C4—431no—
236C23×C4—432no—
237C2×M16—22no—
238C2×M16—862no—
239C4⋊Q8—434no—
240C4⋊Q8—432no—
241C4⋊Q8—862no—
242C4⋊Q8—862no—
243C4⋊Q8C22×D4312yes—
244#125—434no—
245#125—432no—
246#125—864no—
247#125C4×D4312yes—
248C22⋊Q8—22no—
249C22⋊Q8—864no—
250C22⋊Q8—864no—
251C22⋊Q8—862no—
252C22⋊Q8—862no—
253C22⋊Q8—862no—
254C4×Q8—22no—
255C4×Q8—864no—
256C4×Q8—864no—
257C4×Q8—862no—
258C4×Q8—862no—
259C4×Q8—862no—
260#117—432no—
261#117—434no—
262#117—862no—
263#117—862no—
264#117C4×D4312yes—
265C2×C4⋊C4—434no—
266C2×C4⋊C4—864no—
267C2×C4⋊C4—862no—
268C2×C22⋊C4—862no—
269C2×C42—431no—
270C2×C42—432no—
271C2×C42—431no—
272C2×C42—431no—
273C2×C42—432no—
274C2×C42—432no—
275C2×C42—431no—
276C2×C42—431no—
277C2×C42—862no—
278C2×C42C22×D4311yescharacteristic
279C4⋊C8—22no—
280C4⋊C8—434no—
281C4⋊C8—862no—
282C4⋊C8C22×D4314yes—
283C22⋊C8—864no—
284C22⋊C8—864no—
285#97—864no—
286C4×C8—864no—
287C24—431no—
288C2×Q8—22no—
289C2×Q8—864no—
290C2×Q8—862no—
291C2×Q8—1724no—
292C2×Q8—1724no—
293C22×C4—431no—
294C22×C4—431no—
295C22×C4—432no—
296C22×C4—431no—
297C22×C4—431no—
298C22×C4—431no—
299C22×C4—432no—
300C22×C4—432no—
301C22×C4—432no—
302C22×C4—432no—
303C22×C4—864no—
304C22×C4—864no—
305C22×C4—1722no—
306C22×C4—1722no—
307C22×C4C2×C4×D4311yescharacteristic
308C22×C4D4×D4312yes—
309M16—44no—
310M16—1724no—
311C2×C8—24no—
312C2×C8—864no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—434no—
316C4⋊C4—864no—
317C4⋊C4—864no—
318C4⋊C4—864no—
319C4⋊C4—864no—
320C4⋊C4—864no—
321C4⋊C4—862no—
322C4⋊C4C2×C4×D4314yes—
323C22⋊C4—22no—
324C22⋊C4—864no—
325C22⋊C4—862no—
326C42—22no—
327C42—432no—
328C42—432no—
329C42—432no—
330C42—431no—
331C42—431no—
332C42—432no—
333C42—432no—
334C42—864no—
335C42—862no—
336C42C2×C4×D4311yescharacteristic
337C42D4×D4312yes—
338C42C23×D4311yescharacteristic
339C23—431no—
340C23—431no—
341C23—862no—
342C23—1722no—
343C23C22⋊C4×D4311yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—1728no—
347Q8—1722no—
348Q8—1722no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—431no—
354C2×C4—434no—
355C2×C4—431no—
356C2×C4—432no—
357C2×C4—432no—
358C2×C4—862no—
359C2×C4—862no—
360C2×C4—862no—
361C2×C4—862no—
362C2×C4—862no—
363C2×C4—862no—
364C2×C4—862no—
365C2×C4—862no—
366C2×C4—864no—
367C2×C4—1722no—
368C2×C4—1722no—
369C2×C4—1728no—
370C2×C4—1722no—
371C2×C4C22⋊C4×D4312yes—
372C2×C4C22⋊C4×D4311yescharacteristic
373C2×C4C22×C4×D4311yescharacteristic
374C2×C4C2×D4×D4312yes—
375C8—44no—
376C8—1724no—
377C22—21no—
378C22—22no—
379C22—432no—
380C22—862no—
381C22—861no—
382C22—862no—
383C22—1724no—
384C22C2×C22⋊C4×D4311yescharacteristic
385C4—21no—
386C4—22no—
387C4—44no—
388C4—432no—
389C4—861no—
390C4—862no—
391C4—862no—
392C4—1724no—
393C4#101×D4312yes—
394C4#1458369512yes—
395C4#1458369812yes—
396C2—41no—
397C2—862no—
398C2—1721no—
399C2C2×#101×D4311yescharacteristic
400C2#1914311112yes—
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC43×#1561C2
socleC22×C43C22×C22⋊C4
prime radicalC23×D43C22⋊C4
cosocleC42×C43C24
hypercenter#1561D43
hypocenterC43#17739
LCSG ▷ C2×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
3. #19133333C2semidirect
4. #19133332C2semidirect
5. #19133310C2semidirect
6. #19133258C2semidirect
7. C43×#1561C2semidirect
27. #14582646C22semidirect
28. #14582645C22semidirect
29. #14582644C22semidirect
32. C43×#332C22semidirect
200. #1561D43semidirect
232. #332C2×D43semidirect
233. C43#17739semidirect

Maximal subgroups

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

subgroupindex
#533×D432
#191432272
#191333332
#191333322
#191333102
#191332582
C43×#15612
#1773943