Finite Group Databasebeta

← #25070519

Subgroups — #25070519

Order 11008.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#14583530C211yesmaximal, characteristic
2#19145367C212yesmaximal
3#19133636C214yesmaximal, complemented
4#19133568C214yesmaximal, complemented
5#19133166C212yesmaximal
6C2×#14582625C211yesmaximal, characteristic
7C43×#2627C211yesmaximal, characteristic
8#14583530—24no—
9#14583530—22no—
10#14583530C2214yes—
11C2×#13063324—22no—
12C2×#13063316—22no—
13C2×#13063304—21no—
14C2×#13063304C2212yes—
15C2×#13063284—21no—
16#14583234C2214yes—
17#14583232—22no—
18#14583138—24no—
19#14583055C2214yes—
20#14583054—22no—
21C2×C22⋊(C43⋊C8)C2212yes—
22#14582635C2218yes—
23#14582625C2214yes—
24C2×C22⋊Q8×C43C2211yescharacteristic
25C22×C4⋊C4×C43C2211yescharacteristic
26C43×#400C2212yes—
27C43×#398—22no—
28C43×#395C2212yes—
29C43×#385—21no—
30C43×#381—21no—
31C43×#380C2211yescharacteristic
32C22×Dic86—21no—
33C22×Dic86D412yes—
34C2×#10621885—24no—
35C22×C43⋊C8—22no—
36#13063324—28no—
37#13063324—28no—
38#13063316—28no—
39#13063316—28no—
40#13063304—24no—
41#13063304—24no—
42#13063304—22no—
43#13063304—22no—
44#13063304D414yes—
45#13063304C2314yes—
46#13063284—24no—
47#13063284—24no—
48C2×C4⋊Dic43—21no—
49C2×C4⋊Dic43—21no—
50C2×C4⋊Dic43—21no—
51C2×C4⋊Dic43D412yes—
52C2×Dic43⋊C4—22no—
53C2×Dic43⋊C4—22no—
54C2×C22⋊Dic43—22no—
55C2×C4×Dic43—22no—
56#13063258—24no—
57#13063249—24no—
58#13063248—24no—
59C22⋊(C43⋊C8)—22no—
60C22⋊(C43⋊C8)C2314yes—
61C22×Q8×C43C2311yescharacteristic
62C23×C4×C43D411yescharacteristic
63C23×C4×C43D411yescharacteristic
64C23×C4×C43C2311yescharacteristic
65C22⋊Q8×C43—22no—
66C22⋊Q8×C43—22no—
67C2×C4⋊C4×C43—22no—
68C2×C4⋊C4×C43—22no—
69C2×C4⋊C4×C43—21no—
70C2×C4⋊C4×C43—21no—
71C2×C4⋊C4×C43—22no—
72C2×C4⋊C4×C43D412yes—
73C2×C4⋊C4×C43C2311yescharacteristic
74C2×C4⋊C4×C43C2312yes—
75C2×C22⋊C4×C43—21no—
76C2×C22⋊C4×C43—21no—
77#95×C43—22no—
78#95×C43—22no—
79#95×C43—22no—
80#95×C43C2312yes—
81C24×C43C2×D411yescharacteristic
82C2×Q8×C43—22no—
83C2×Q8×C43—81no—
84C22×C4×C43—21no—
85C22×C4×C43—21no—
86C22×C4×C43—21no—
87C22×C4×C43—21no—
88C22×C4×C43—22no—
89C22×C4×C43—21no—
90C22×C4×C43—22no—
91C22×C4×C43—21no—
92C22×C4×C43—81no—
93C22×C4×C43—81no—
94C22×C4×C43Q1614yescomplemented
95C22×C4×C43C2×D412yes—
96C22×C4×C43C2×D412yes—
97C22×C4×C43C2×D412yes—
98C22×C4×C43C2×D412yes—
99C22×C4×C43C2×D411yescharacteristic
100C22×C4×C43C2411yescharacteristic
101C22×Dic43—22no—
102C22×Dic43—22no—
103C2×Dic86—24no—
104C2×Dic86—22no—
105C2×Dic86—82no—
106C2×Dic86—82no—
107C2×Dic86C2×D414yes—
108C4⋊C4×C43—24no—
109C4⋊C4×C43—24no—
110C4⋊C4×C43—22no—
111C4⋊C4×C43—42no—
112C4⋊C4×C43—81no—
113C4⋊C4×C43—82no—
114C22⋊C4×C43—41no—
115C22⋊C4×C43—41no—
116C22⋊C4×C43—42no—
117C22⋊Dic43—24no—
118C22⋊Dic43—24no—
119#10621885—84no—
120C4⋊Dic43—24no—
121C4⋊Dic43—24no—
122C4⋊Dic43—24no—
123C4⋊Dic43—22no—
124C4⋊Dic43C2×D414yes—
125Dic43⋊C4—28no—
126Dic43⋊C4—24no—
127Dic43⋊C4—24no—
128C4×Dic43—28no—
129C2×C43⋊C8—24no—
130C2×C43⋊C8—82no—
131C23×C43—41no—
132C23×C43—41no—
133C23×C43#12011yescharacteristic
134C23×C43#12011yescharacteristic
135C23×C43C2×Q1612yes—
136C23×C43#13712yes—
137C23×C43C22×D411yescharacteristic
138Q8×C43—44no—
139C2×C4×C43—22no—
140C2×C4×C43—22no—
141C2×C4×C43—24no—
142C2×C4×C43—24no—
143C2×C4×C43—22no—
144C2×C4×C43—42no—
145C2×C4×C43—41no—
146C2×C4×C43—42no—
147C2×C4×C43—41no—
148C2×C4×C43—42no—
149C2×C4×C43—42no—
150C2×C4×C43—42no—
151C2×C4×C43—42no—
152C2×C4×C43—42no—
153C2×C4×C43—42no—
154C2×C4×C43—41no—
155C2×C4×C43—41no—
156C2×C4×C43—42no—
157C2×C4×C43—81no—
158C2×C4×C43#12012yes—
159C2×C4×C43C2×Q1614yes—
160C2×C4×C43C22×D412yes—
161C2×Dic43—24no—
162C2×Dic43—24no—
163C2×Dic43—44no—
164C2×Dic43—44no—
165Dic86—44no—
166Dic86—82no—
167Dic86—84no—
168C43⋊C8—44no—
169#18295—431nomaximal, Sylow
170C22×C43—21no—
171C22×C43—21no—
172C22×C43—21no—
173C22×C43—21no—
174C22×C43—21no—
175C22×C43—21no—
176C22×C43—42no—
177C22×C43—42no—
178C22×C43#45112yes—
179C22×C43#45112yes—
180C22×C43C2×#12011yescharacteristic
181C22×C43C22×Q1611yescharacteristic
182C22×C43C2×#13711yescharacteristic
183C4×C43—24no—
184C4×C43—22no—
185C4×C43—22no—
186C4×C43—42no—
187C4×C43—82no—
188C4×C43—81no—
189C4×C43—81no—
190Dic43—44no—
191Dic43—44no—
192C2×#544—431no—
193#3189—432no—
194#2627D4311yescharacteristic
195#1617—434no—
196#1605—434no—
197#1520—432no—
198C2×#324—431no—
199C2×C43—21no—
200C2×C43—21no—
201C2×C43—22no—
202C2×C43—22no—
203C2×C43C2×#45111yescharacteristic
204C2×C43#302011yescharacteristic
205C2×C43#302611yescharacteristic
206#544—434no—
207#544—864no—
208#544—862no—
209C2×C4⋊Q8—862no—
210C2×#125—862no—
211C2×C22⋊Q8—432no—
212C2×C22⋊Q8—861no—
213C2×C22⋊Q8C2×D4311yescharacteristic
214C2×#117—861no—
215C22×C4⋊C4C2×D4311yescharacteristic
216#484—434no—
217#484—862no—
218#467—864no—
219#451—434no—
220#451—862no—
221C2×C22⋊C8—432no—
222#400C2×D4312yes—
223#398—22no—
224#395C2×D4312yes—
225#385—21no—
226#381—21no—
227#380C2×D4311yescharacteristic
228#328—438no—
229#324—434no—
230C43#1829511yescharacteristic, complemented, Sylow
231C22×Q8—432no—
232C22×Q8—861no—
233C22×Q8C22×D4311yescharacteristic
234C23×C4C22⋊D4311yescharacteristic
235C23×C4C22⋊D4311yescharacteristic
236C23×C4C22×D4311yescharacteristic
237C2×Q16—864no—
238C22×C8—862no—
239C4⋊Q8—868no—
240C4⋊Q8—868no—
241#125—868no—
242#125—868no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—434no—
246C22⋊Q8—434no—
247C22⋊Q8—864no—
248C22⋊Q8—862no—
249C22⋊Q8—862no—
250C22⋊Q8—864no—
251#117—864no—
252#117—864no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—861no—
260C2×C4⋊C4—861no—
261C2×C4⋊C4—861no—
262C2×C4⋊C4—862no—
263C2×C4⋊C4—862no—
264C2×C4⋊C4C22⋊D4312yes—
265C2×C4⋊C4C22×D4311yescharacteristic
266C2×C4⋊C4C22×D4312yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—862no—
270C2×C42—862no—
271#107—864no—
272#103—864no—
273#103—864no—
274C22⋊C8—434no—
275C22⋊C8—862no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95C22×D4312yes—
280C24C2×C22⋊D4311yescharacteristic
281C2×Q8—22no—
282C2×Q8—81no—
283C2×Q8—434no—
284C2×Q8—864no—
285C2×Q8—862no—
286C2×Q8—3442no—
287C2×Q8—3442no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—862no—
299C22×C4—862no—
300C22×C4#1062188514yescomplemented
301C22×C4D4×D4312yes—
302C22×C4D4×D4312yes—
303C22×C4C2×C22⋊D4312yes—
304C22×C4C2×C22⋊D4312yes—
305C22×C4C2×C22⋊D4311yescharacteristic
306C22×C4C23×D4311yescharacteristic
307Q16—3444no—
308C2×C8—864no—
309C2×C8—3442no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C4⋊C4—434no—
317C4⋊C4—864no—
318C4⋊C4—864no—
319C4⋊C4—864no—
320C4⋊C4—868no—
321C4⋊C4—862no—
322C4⋊C4—864no—
323C4⋊C4—864no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—864no—
328C22⋊C4—864no—
329C42—868no—
330C23—41no—
331C23—41no—
332C23#1306329311yescharacteristic
333C23#1306329311yescharacteristic
334C23C2×#1062188512yes—
335C23#1306335512yes—
336C23C22×C22⋊D4311yescharacteristic
337Q8—44no—
338Q8—3442no—
339Q8—3444no—
340Q8—1724no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—864no—
361C2×C4—864no—
362C2×C4—1724no—
363C2×C4—1724no—
364C2×C4#1306329312yes—
365C2×C4C2×#1062188514yes—
366C2×C4C2×D4×D4312yes—
367C8—1724no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1458305412yes—
377C22#1458305412yes—
378C22C2×#1306329311yescharacteristic
379C22C22×#1062188511yescharacteristic
380C22C2×#1306335511yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—1724no—
389C4—1724no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1458305411yescharacteristic
395C2#1914395811yescharacteristic
396C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2627C2
socleC22×C43C2×#120
prime radicalC24×C43C2×D4
hypercenter#2627D43
hypocenterC43#18295
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #19133636C2semidirect
4. #19133568C2semidirect
94. C22×C4×C43Q16semidirect
230. C43#18295semidirect
300. C22×C4#10621885semidirect

Maximal subgroups

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

subgroupindex
C2×#145835302
#191453672
#191336362
#191335682
#191331662
C2×#145826252
C43×#26272
#1829543