Finite Group Databasebeta

← #24989119

Subgroups — #24989119

Order 11008.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14583520C211yesmaximal, characteristic
2#19133313C212yesmaximal, complemented
3#19133218C212yesmaximal, complemented
4#19132929C211yesmaximal, characteristic, complemented
5C43×#1464C211yesmaximal, characteristic, complemented
6#14583520—24no—
7#14583520—24no—
8C2×#13063325C2211yescharacteristic
9C2×#13063320—21no—
10C2×#13063313C2211yescharacteristic
11C2×#13063311—21no—
12C2×#13063307—21no—
13C2×#13063304C411yescharacteristic
14C2×#13063303—21no—
15C2×#13063299C411yescharacteristic
16C2×C4×Dic86C411yescharacteristic
17C2×C4×D172C411yescharacteristic
18#14582956C2212yes—
19#14582646—24no—
20#14582638—24no—
21#14582625—21no—
22#14582623—21no—
23C2×#117×C43C2211yescharacteristic
24C43×#432C2212yes—
25C43×#324—21no—
26C43×#323—21no—
27C22×Dic86—21no—
28C22×Dic86C2×C411yescharacteristic
29C22×D172C2×C411yescharacteristic
30C22×C22⋊D43—21no—
31C22×C4×D43—21no—
32#13063325—22no—
33#13063325D414yes—
34#13063320—28no—
35#13063313—22no—
36#13063313D414yes—
37#13063311—28no—
38#13063307—28no—
39#13063304—22no—
40#13063304—22no—
41#13063303—28no—
42#13063299—22no—
43#13063299—22no—
44C4×Dic86—22no—
45C4×Dic86—22no—
46C4×D172—22no—
47C4×D172—22no—
48C2×C4⋊Dic43—21no—
49C2×C4⋊Dic43C2×C411yescharacteristic
50C2×C4⋊Dic43C2×C411yescharacteristic
51C2×Dic43⋊C4—21no—
52C2×Dic43⋊C4—21no—
53C2×C22⋊Dic43—21no—
54C2×(C2×D43)⋊C4—21no—
55C2×(C2×D43)⋊C4—21no—
56C2×(C2×D43)⋊C4—21no—
57C2×(C2×D43)⋊C4—21no—
58C2×C4×Dic43—21no—
59C4⋊(C43⋊C8)—22no—
60C22⋊(C43⋊C8)—22no—
61#13063236—22no—
62C23×C4×C43C2×C411yescharacteristic
63#117×C43—22no—
64#117×C43—22no—
65C2×C4⋊C4×C43D411yescharacteristic
66C2×C4⋊C4×C43D411yescharacteristic
67C2×C22⋊C4×C43D411yescharacteristic
68C2×C22⋊C4×C43D411yescharacteristic
69C2×C42×C43C2×C411yescharacteristic
70C2×C42×C43C2311yescharacteristic
71C4⋊C8×C43—22no—
72C22⋊C8×C43—22no—
73#97×C43—22no—
74C24×C43C22⋊C411yescharacteristic
75C23×D43—21no—
76C22×C4×C43—21no—
77C22×C4×C43—21no—
78C22×C4×C43—42no—
79C22×C4×C43C22⋊C411yescharacteristic
80C22×C4×C43C22⋊C411yescharacteristic
81C22×C4×C43C22⋊C411yescharacteristic
82C22×C4×C43C22×C411yescharacteristic
83C22×C4×C43C2×D411yescharacteristic
84C22×C4×C43C2×D411yescharacteristic
85C2×C22⋊D43—24no—
86C2×C22⋊D43—82no—
87C22×Dic43—21no—
88C22×Dic43—21no—
89C22×Dic43—21no—
90C2×D172—21no—
91C2×D172—42no—
92C2×D172C22⋊C412yes—
93C2×C4×D43—24no—
94C2×C4×D43—82no—
95C2×Dic86—21no—
96C2×Dic86—24no—
97C2×Dic86—42no—
98C2×Dic86—82no—
99C2×Dic86C22⋊C412yes—
100C2×C8×C43—42no—
101C4⋊C4×C43—22no—
102C4⋊C4×C43—22no—
103C22⋊C4×C43—22no—
104C22⋊C4×C43—22no—
105C42×C43—21no—
106C42×C43—21no—
107C42×C43C2×D412yes—
108C42×C43C2×D412yes—
109C22⋊Dic43—24no—
110(C2×D43)⋊C4—24no—
111(C2×D43)⋊C4—24no—
112(C2×D43)⋊C4—24no—
113(C2×D43)⋊C4—24no—
114C4⋊Dic43—21no—
115C4⋊Dic43—21no—
116C4⋊Dic43—24no—
117C4⋊Dic43C22⋊C412yes—
118C4⋊Dic43C22⋊C412yes—
119Dic43⋊C4—24no—
120Dic43⋊C4—24no—
121C4×Dic43—24no—
122C2×C43⋊C8—42no—
123C23×C43—21no—
124C23×C43—22no—
125C23×C43—41no—
126C23×C43—41no—
127C23×C43C2×C22⋊C411yescharacteristic
128C22×D43—22no—
129C22×D43—42no—
130C22×D43—42no—
131C22×D43—42no—
132C2×C4×C43—22no—
133C2×C4×C43—21no—
134C2×C4×C43—21no—
135C2×C4×C43—21no—
136C2×C4×C43—22no—
137C2×C4×C43—41no—
138C2×C4×C43—41no—
139C2×C4×C43—42no—
140C2×C4×C43—42no—
141C2×C4×C43—41no—
142C2×C4×C43—41no—
143C2×C4×C43C2×C22⋊C411yescharacteristic
144C2×C4×C43C2×C22⋊C411yescharacteristic
145C2×C4×C43#12012yes—
146C2×C4×C43#12012yes—
147C22⋊D43—84no—
148C2×Dic43—22no—
149C2×Dic43—22no—
150C2×Dic43—22no—
151C2×Dic43—42no—
152C2×Dic43—42no—
153C2×Dic43—42no—
154D172—42no—
155D172—42no—
156C4×D43—84no—
157Dic86—42no—
158Dic86—42no—
159Dic86—84no—
160C8×C43—82no—
161C43⋊C8—82no—
162#8523—431nomaximal, Sylow
163C22×C43—21no—
164C22×C43—21no—
165C22×C43—41no—
166C22×C43—41no—
167C22×C43—41no—
168C22×C43—82no—
169C22×C43#37911yescharacteristic
170C22×C43#41312yes—
171C2×D43—42no—
172C2×D43—42no—
173C2×D43—42no—
174C2×D43—84no—
175C4×C43—22no—
176C4×C43—22no—
177C4×C43—21no—
178C4×C43—21no—
179C4×C43—41no—
180C4×C43—41no—
181C4×C43—41no—
182Dic43—42no—
183Dic43—42no—
184Dic43—42no—
185C2×#542—431no—
186#1556—432no—
187#1533—432no—
188#1464—431no—
189#1464D4311yescharacteristic, complemented
190C2×C43—22no—
191C2×C43—81no—
192C2×C43#178511yescharacteristic
193C2×C43#179511yescharacteristic
194C2×C43#179611yescharacteristic
195D43—82no—
196#542—864no—
197#542—864no—
198C2×C4⋊Q8—431no—
199C2×#126—861no—
200C2×#124—431no—
201C2×#124—861no—
202C2×#123—861no—
203C2×C22⋊Q8—431no—
204C2×C22⋊Q8—861no—
205C2×#121—431no—
206C2×C4×Q8—431no—
207C2×C4×D4—431no—
208C2×#117C2×D4311yescharacteristic
209#432—432no—
210#432C2×D4312yes—
211#332—864no—
212#329—864no—
213#324—21no—
214#324—861no—
215#323—21no—
216#323—861no—
217C43#852311yescharacteristic, complemented, Sylow
218C22×Q8—431no—
219C22×Q8—861no—
220C22×D4—431no—
221C22×D4—861no—
222C23×C4—861no—
223C23×C4C4×D4311yescharacteristic
224C4⋊Q8—434no—
225C4⋊Q8—862no—
226#126—868no—
227#124—434no—
228#124—868no—
229#124—862no—
230#123—868no—
231C22⋊Q8—868no—
232C22⋊Q8—862no—
233C22⋊Q8—862no—
234#121—862no—
235#121—862no—
236C4×Q8—862no—
237C4×Q8—862no—
238C4×D4—862no—
239C4×D4—862no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—431no—
243C2×C4⋊C4—431no—
244C2×C4⋊C4—861no—
245C2×C4⋊C4—861no—
246C2×C4⋊C4—861no—
247C2×C4⋊C4D17211yescharacteristic
248C2×C4⋊C4C22⋊D4311yescharacteristic
249C2×C22⋊C4—861no—
250C2×C22⋊C4—861no—
251C2×C22⋊C4—861no—
252C2×C22⋊C4—861no—
253C2×C22⋊C4—861no—
254C2×C22⋊C4D17211yescharacteristic
255C2×C22⋊C4C22⋊D4311yescharacteristic
256C2×C42—861no—
257C2×C42C4×D4311yescharacteristic
258C2×C42C22×D4311yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—862no—
261C22⋊C8—22no—
262C22⋊C8—862no—
263#97—22no—
264#97—862no—
265C24—861no—
266C24(C2×D43)⋊C411yescharacteristic
267C2×Q8—432no—
268C2×Q8—861no—
269C2×Q8—864no—
270C2×Q8—3442no—
271C2×Q8—1722no—
272C2×D4—432no—
273C2×D4—864no—
274C2×D4—861no—
275C2×D4—3442no—
276C2×D4—1722no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—864no—
281C22×C4—861no—
282C22×C4—861no—
283C22×C4—861no—
284C22×C4—3442no—
285C22×C4(C2×D43)⋊C411yescharacteristic
286C22×C4(C2×D43)⋊C411yescharacteristic
287C22×C4(C2×D43)⋊C411yescharacteristic
288C22×C4C2×C4×D4311yescharacteristic
289C22×C4C2×D17211yescharacteristic
290C22×C4C2×C22⋊D4311yescharacteristic
291C2×C8—42no—
292C2×C8—1722no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—432no—
296C4⋊C4—432no—
297C4⋊C4—864no—
298C4⋊C4—864no—
299C4⋊C4—861no—
300C4⋊C4—861no—
301C4⋊C4—864no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—864no—
305C22⋊C4—864no—
306C22⋊C4—864no—
307C22⋊C4—864no—
308C22⋊C4—864no—
309C42—21no—
310C42—21no—
311C42—864no—
312C42D4×D4312yes—
313C42D4×D4312yes—
314C23—21no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—862no—
319C23—1722no—
320C23—1722no—
321C23—1722no—
322C23C2×(C2×D43)⋊C411yescharacteristic
323Q8—3444no—
324Q8—1722no—
325Q8—1722no—
326D4—3444no—
327D4—1722no—
328D4—1722no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—862no—
341C2×C4—862no—
342C2×C4—862no—
343C2×C4—3444no—
344C2×C4—1722no—
345C2×C4—1722no—
346C2×C4—1722no—
347C2×C4C22⋊C4×D4311yescharacteristic
348C2×C4C22⋊C4×D4311yescharacteristic
349C2×C4#1306329312yes—
350C2×C4#1306329512yes—
351C8—82no—
352C8—3442no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—82no—
359C22—3444no—
360C22—1722no—
361C22—1722no—
362C22—1722no—
363C22#1458275411yescharacteristic
364C22#1458288112yes—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—1722no—
373C4—1722no—
374C4—1722no—
375C2—22no—
376C2—81no—
377C2—3442no—
378C2#1913452111yescharacteristic
379C2#1913456311yescharacteristic
380C2#1913457111yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1464C2
socleC22×C43#379
prime radicalC2×#13063299C4
cosocleC2×C42×C43C23
hypercenter#1464D43
hypocenterC43#8523
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #19133313C2semidirect
3. #19133218C2semidirect
4. #19132929C2semidirect
5. C43×#1464C2semidirect
189. #1464D43semidirect
217. C43#8523semidirect

Maximal subgroups

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

subgroupindex
C2×#145835202
#191333132
#191332182
#191329292
C43×#14642
#852343