Finite Group Databasebeta

← #24993762

Subgroups — #24993762

Order 11008.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#19144539C211yesmaximal, characteristic
2#19137486C211yesmaximal, characteristic
3#19135581C211yesmaximal, characteristic
4#19134692C211yesmaximal, characteristic
5#19132502C211yesmaximal, characteristic
6#19132481C211yesmaximal, characteristic
7C43×#1508C211yesmaximal, characteristic
8C2×#13063284C2211yescharacteristic
9#14583237—22no—
10#14583232—21no—
11#14583227—21no—
12#14582939C412yescomplemented
13#14582908C412yescomplemented
14#14582863C2211yescharacteristic
15#14582797—21no—
16#14582793—22no—
17#14582766—21no—
18#14582759C2211yescharacteristic
19C4×C4⋊Dic43—21no—
20#14582750—21no—
21C4×C22⋊Dic43C2211yescharacteristic
22#14582666—22no—
23#14582660—22no—
24C2×C22⋊Q8×C43C2211yescharacteristic
25C43×#407C2211yescharacteristic
26C4×C22⋊C4×C43C2211yescharacteristic
27C43×#328—21no—
28C43×#328—21no—
29C2×#10621877C2×C412yes—
30#13063284—21no—
31#13063284—21no—
32#13063284C2×C412yes—
33#13063284D412yes—
34C2×C4⋊Dic43—21no—
35C2×C4⋊Dic43D411yescharacteristic
36C2×C4⋊Dic43D411yescharacteristic
37C2×Dic43⋊C4—22no—
38C2×C22⋊Dic43—21no—
39C2×C22⋊Dic43—21no—
40C2×C22⋊Dic43D412yes—
41C2×C4×Dic43—21no—
42C2×C4×Dic43—21no—
43C2×C4×Dic43—22no—
44#13063258—22no—
45#13063256—22no—
46#13063249—22no—
47#13063249—22no—
48C22⋊(C43⋊C8)—21no—
49C22⋊(C43⋊C8)—21no—
50C43⋊#95—22no—
51C43⋊#95—22no—
52C43⋊#95—21no—
53C43⋊#95—21no—
54C22×Q8×C43C2×C411yescharacteristic
55C23×C4×C43C2311yescharacteristic
56C2×M16×C43—21no—
57C22⋊Q8×C43—21no—
58C22⋊Q8×C43—21no—
59C22⋊Q8×C43—21no—
60C22⋊Q8×C43—21no—
61C2×C4⋊C4×C43—21no—
62C2×C4⋊C4×C43C2×C411yescharacteristic
63C2×C22⋊C4×C43—21no—
64C2×C22⋊C4×C43—21no—
65C2×C42×C43—21no—
66C22⋊C8×C43—21no—
67C22⋊C8×C43—21no—
68#95×C43D411yescharacteristic
69#95×C43D411yescharacteristic
70C24×C43C2×D411yescharacteristic
71C2×Q8×C43—21no—
72C2×Q8×C43—21no—
73C2×Q8×C43—42no—
74C22×C4×C43—21no—
75C22×C4×C43—21no—
76C22×C4×C43—21no—
77C22×C4×C43—21no—
78C22×C4×C43—42no—
79C22×C4×C43C22×C411yescharacteristic
80C22×C4×C43C2×D411yescharacteristic
81C22×C4×C43C2×D411yescharacteristic
82C22×C4×C43C2×D411yescharacteristic
83C22×C4×C43#4111yescharacteristic
84C22×C4×C43#4111yescharacteristic
85C22×Dic43—21no—
86C22×Dic43—21no—
87C22×Dic43—22no—
88C22×Dic43—41no—
89M16×C43—42no—
90C2×C8×C43—41no—
91C4⋊C4×C43—21no—
92C4⋊C4×C43—21no—
93C4⋊C4×C43—41no—
94C4⋊C4×C43—41no—
95C22⋊C4×C43—42no—
96C22⋊C4×C43—41no—
97C22⋊C4×C43—41no—
98C42×C43—42no—
99C22⋊Dic43—22no—
100C22⋊Dic43—22no—
101C22⋊Dic43—22no—
102C22⋊Dic43—42no—
103C22⋊Dic43—41no—
104C4⋊Dic43—21no—
105C4⋊Dic43—21no—
106C4⋊Dic43—42no—
107C4⋊Dic43C22⋊C412yes—
108C4⋊Dic43C22⋊C412yes—
109Dic43⋊C4—44no—
110C4×Dic43—22no—
111C4×Dic43—22no—
112C4×Dic43—42no—
113C4×Dic43—82no—
114#10621877—42no—
115C2×C43⋊C8—22no—
116C23×C43—21no—
117C23×C43—21no—
118C23×C43—41no—
119C23×C43—41no—
120C23×C43C4×D411yescharacteristic
121C23×C43#12011yescharacteristic
122C23×C43#12111yescharacteristic
123Q8×C43—42no—
124Q8×C43—81no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—22no—
129C2×C4×C43—41no—
130C2×C4×C43—41no—
131C2×C4×C43—41no—
132C2×C4×C43—41no—
133C2×C4×C43—41no—
134C2×C4×C43—41no—
135C2×C4×C43—41no—
136C2×C4×C43—41no—
137C2×C4×C43—41no—
138C2×C4×C43—41no—
139C2×C4×C43—81no—
140C2×C4×C43—81no—
141C2×C4×C43C2×C22⋊C411yescharacteristic
142C2×C4×C43C4×D411yescharacteristic
143C2×C4×C43#12111yescharacteristic
144C2×C4×C43#12311yescharacteristic
145C2×Dic43—22no—
146C2×Dic43—22no—
147C2×Dic43—42no—
148C2×Dic43—41no—
149C2×Dic43—41no—
150C2×Dic43—42no—
151C2×Dic43—42no—
152C2×Dic43—42no—
153C2×Dic43—42no—
154C2×Dic43—81no—
155C2×Dic43—81no—
156C8×C43—81no—
157C43⋊C8—42no—
158#9362—431nomaximal, Sylow
159C22×C43—21no—
160C22×C43—21no—
161C22×C43—21no—
162C22×C43—21no—
163C22×C43—21no—
164C22×C43—21no—
165C22×C43—21no—
166C22×C43—21no—
167C22×C43—81no—
168C22×C43—81no—
169C22×C43#38611yescharacteristic
170C22×C43#45511yescharacteristic
171C22×C43#45611yescharacteristic
172C4×C43—21no—
173C4×C43—21no—
174C4×C43—21no—
175C4×C43—42no—
176C4×C43—81no—
177C4×C43—81no—
178C4×C43—81no—
179C4×C43#41012yes—
180Dic43—42no—
181Dic43—42no—
182Dic43—82no—
183Dic43—82no—
184#3095—431no—
185#2347—431no—
186#1996—431no—
187#1826—431no—
188#1508D4311yescharacteristic
189#1305—431no—
190#1297—431no—
191C2×C43—21no—
192C2×C43—21no—
193C2×C43—41no—
194C2×C43—41no—
195C2×C43#188211yescharacteristic
196C2×C43#189911yescharacteristic
197C2×C43#190611yescharacteristic
198C2×C22⋊Q8C2×D4311yescharacteristic
199C2×#117—431no—
200#485—862no—
201#484—861no—
202#483—861no—
203#428—432no—
204#419—432no—
205#407—431no—
206#407C2×D4311yescharacteristic
207#389—861no—
208#388—862no—
209#383—861no—
210#380—431no—
211C4×C4⋊C4—861no—
212C4×C4⋊C4—861no—
213C4×C22⋊C4—431no—
214C4×C22⋊C4C2×D4311yescharacteristic
215#342—862no—
216#339—862no—
217#328—21no—
218#328—21no—
219C43#936211yescharacteristic, complemented, Sylow
220C22×Q8C4×D4311yescharacteristic
221C23×C4C22×D4311yescharacteristic
222C2×M16—21no—
223C2×M16—432no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#117—432no—
229#117—432no—
230#117—861no—
231#117—861no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—431no—
234C2×C4⋊C4—431no—
235C2×C4⋊C4—862no—
236C2×C4⋊C4—861no—
237C2×C4⋊C4C4×D4311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—432no—
241C2×C22⋊C4—861no—
242C2×C22⋊C4—861no—
243C2×C42—21no—
244C2×C42—861no—
245C2×C42—862no—
246C2×C42—861no—
247#107—862no—
248#106—862no—
249#103—862no—
250#103—862no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—861no—
254C22⋊C8—861no—
255#95—862no—
256#95—862no—
257#95—861no—
258#95—861no—
259#95C22⋊D4311yescharacteristic
260#95C22⋊D4311yescharacteristic
261C24D4×D4311yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—861no—
271C22×C4—861no—
272C22×C4—862no—
273C22×C4—1721no—
274C22×C4C2×C4×D4311yescharacteristic
275C22×C4#1062189811yescharacteristic
276C22×C4D4×D4311yescharacteristic
277C22×C4D4×D4311yescharacteristic
278C22×C4#1062190011yescharacteristic
279C22×C4C2×C22⋊D4311yescharacteristic
280M16—42no—
281M16—1722no—
282C2×C8—41no—
283C2×C8—862no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—432no—
289C4⋊C4—432no—
290C4⋊C4—861no—
291C4⋊C4—861no—
292C4⋊C4—1724no—
293C4⋊C4—1722no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—862no—
298C22⋊C4—862no—
299C22⋊C4—862no—
300C22⋊C4—1721no—
301C22⋊C4—1722no—
302C42—42no—
303C42—862no—
304C42—862no—
305C42—3442no—
306C42—1722no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23(C22⋊D43)⋊C411yescharacteristic
312C23#1306329311yescharacteristic
313C23#1306329811yescharacteristic
314Q8—42no—
315Q8—81no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—862no—
333C2×C4—862no—
334C2×C4—3441no—
335C2×C4—3441no—
336C2×C4—1722no—
337C2×C4—1722no—
338C2×C4—1722no—
339C2×C4—1722no—
340C2×C4—1722no—
341C2×C4—1721no—
342C2×C4—1721no—
343C2×C4C22⋊C4×D4311yescharacteristic
344C2×C4C4×C22⋊D4311yescharacteristic
345C2×C4#1306329611yescharacteristic
346C2×C4#1306330711yescharacteristic
347C8—81no—
348C8—1722no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1458277511yescharacteristic
360C22#1458307511yescharacteristic
361C22#1458308011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—3442no—
370C4—3442no—
371C4—1722no—
372C4—1722no—
373C4#1458287112yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#1913496111yescharacteristic
379C2#1913505011yescharacteristic
380C2#1913509911yescharacteristic
subgroupG/H
centerC22#14582775
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1508C2
socleC22×C43#386
prime radicalC24×C43C2×D4
cosocleC23×C4×C43C23
hypercenter#1508D43
hypocenterC43#9362
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
12. #14582939C4semidirect
13. #14582908C4semidirect
219. C43#9362semidirect

Maximal subgroups

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

subgroupindex
#191445392
#191374862
#191355812
#191346922
#191325022
#191324812
C43×#15082
#936243