Finite Group Databasebeta

← #24988318

Subgroups — #24988318

Order 11008.

998 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14583553C211yesmaximal, characteristic
2#19134587C211yesmaximal, characteristic, complemented
3#19133266C212yesmaximal, complemented
4#19133021C212yesmaximal, complemented
5C43×#1460C211yesmaximal, characteristic, complemented
6C22×#10621900—21no—
7#14583553—24no—
8#14583553—24no—
9C2×#13063325C2211yescharacteristic
10C2×#13063309—22no—
11C2×#13063305—21no—
12C2×#13063299C412yes—
13C2×C4×D172C2211yescharacteristic
14C22×C4⋊Dic43C412yes—
15#14582923C2212yes—
16C2×#13063242—21no—
17#14582668—21no—
18#14582644—24no—
19#14582628—24no—
20C2×C4×D4×C43C2211yescharacteristic
21C43×#431C2212yes—
22C43×#323—22no—
23C2×#10621900—28no—
24C2×#10621900—44no—
25C2×#10621900—42no—
26C2×#10621900—42no—
27C22×Dic86—21no—
28C22×D172C2×C411yescharacteristic
29C22×C22⋊D43—22no—
30C23×Dic43—22no—
31C22×C4×D43—21no—
32C2×#10621877—22no—
33#13063325—22no—
34#13063325D414yes—
35#13063309—216no—
36#13063305—28no—
37#13063299—24no—
38#13063299—24no—
39C4×D172—22no—
40C4×D172D414yes—
41C2×C4⋊Dic43—22no—
42C2×C4⋊Dic43—21no—
43C2×C4⋊Dic43—24no—
44C2×C4⋊Dic43—24no—
45C2×C4⋊Dic43C2×C411yescharacteristic
46C2×C4⋊Dic43C2×C412yes—
47C2×(C2×D43)⋊C4—22no—
48C2×(C2×D43)⋊C4—21no—
49C4⋊(C43⋊C8)—22no—
50C4⋊(C43⋊C8)—22no—
51#13063242—42no—
52C22×D4×C43D411yescharacteristic
53C23×C4×C43C2×C412yes—
54C4×D4×C43—22no—
55C4×D4×C43—22no—
56C2×C4⋊C4×C43D411yescharacteristic
57C2×C22⋊C4×C43D412yes—
58C2×C42×C43C2311yescharacteristic
59C4⋊C8×C43—22no—
60C22⋊C8×C43—22no—
61C4×C8×C43—22no—
62C24×C43C22⋊C412yes—
63C23×D43—21no—
64C2×D4×C43—22no—
65C2×D4×C43—22no—
66C2×D4×C43D812yes—
67C2×D4×C43SD1612yes—
68C22×C4×C43—22no—
69C22×C4×C43—22no—
70C22×C4×C43—44no—
71C22×C4×C43C22⋊C412yes—
72C22×C4×C43C22×C411yescharacteristic
73C22×C4×C43C2×D411yescharacteristic
74C22×C4×C43C2×D411yescharacteristic
75C2×C22⋊D43—28no—
76C2×C22⋊D43—44no—
77C2×C22⋊D43—44no—
78C22×Dic43—28no—
79C22×Dic43—21no—
80C22×Dic43—22no—
81C22×Dic43—44no—
82C22×Dic43—44no—
83#10621900—416no—
84#10621900—416no—
85C2×D172—21no—
86C2×D172—42no—
87C2×D172C22⋊C412yes—
88C2×C4×D43—24no—
89C2×C4×D43—41no—
90C2×C4×D43—41no—
91C2×C4×D43—42no—
92C2×Dic86—24no—
93C2×Dic86—41no—
94C2×Dic86—41no—
95C2×Dic86—42no—
96C2×C8×C43—42no—
97C4⋊C4×C43—22no—
98C22⋊C4×C43—24no—
99C42×C43C2×D412yes—
100C42×C43C2×D412yes—
101(C2×D43)⋊C4—28no—
102(C2×D43)⋊C4—24no—
103C4⋊Dic43—28no—
104C4⋊Dic43—21no—
105C4⋊Dic43—22no—
106C4⋊Dic43—24no—
107C4⋊Dic43C22⋊C412yes—
108C4⋊Dic43C22⋊C414yes—
109#10621877—42no—
110#10621877—42no—
111C2×C43⋊C8—42no—
112C23×C43—22no—
113C23×C43—22no—
114C23×C43—42no—
115C23×C43—42no—
116C23×C43#10214yes—
117C23×C43C2×C22⋊C411yescharacteristic
118C22×D43—22no—
119C22×D43—42no—
120C22×D43—42no—
121C22×D43—41no—
122C22×D43—41no—
123D4×C43—24no—
124D4×C43—24no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—41no—
128C2×C4×C43—44no—
129C2×C4×C43—44no—
130C2×C4×C43—42no—
131C2×C4×C43C2×C22⋊C411yescharacteristic
132C2×C4×C43C2×C22⋊C411yescharacteristic
133C2×C4×C43#12012yes—
134C2×C4×C43#12012yes—
135C2×C4×C43C2×D811yescharacteristic
136C2×C4×C43C2×SD1611yescharacteristic
137C2×C4×C43AGL(1,ℤ/8)12yes—
138C22⋊D43—416no—
139C2×Dic43—24no—
140C2×Dic43—22no—
141C2×Dic43—416no—
142C2×Dic43—42no—
143C2×Dic43—41no—
144C2×Dic43—42no—
145C2×Dic43—41no—
146D172—24no—
147D172—42no—
148C4×D43—44no—
149C4×D43—44no—
150Dic86—44no—
151Dic86—44no—
152C8×C43—82no—
153C43⋊C8—82no—
154#8387—431nomaximal, Sylow
155C22×C43—21no—
156C22×C43—21no—
157C22×C43—24no—
158C22×C43—44no—
159C22×C43—44no—
160C22×C43—42no—
161C22×C43—42no—
162C22×C43#37911yescharacteristic
163C22×C43C2×#10211yescharacteristic
164C22×C43#41811yescharacteristic
165C2×D43—42no—
166C2×D43—42no—
167C2×D43—42no—
168C2×D43—82no—
169C2×D43—82no—
170C4×C43—22no—
171C4×C43—21no—
172C4×C43—42no—
173C4×C43#41312yes—
174C4×C43#44712yes—
175C4×C43#45012yes—
176Dic43—44no—
177Dic43—42no—
178C2×#547—431no—
179#1800—431no—
180#1546—432no—
181#1486—432no—
182#1460D4311yescharacteristic, complemented
183C2×C43—21no—
184C2×C43—21no—
185C2×C43—44no—
186C2×C43#178511yescharacteristic
187C2×C43#178611yescharacteristic
188C2×C43#179611yescharacteristic
189D43—82no—
190C22×#41—861no—
191#547—864no—
192#547—864no—
193C2×C4⋊Q8—431no—
194C2×#123—862no—
195C2×C22⋊Q8—861no—
196C2×#121—432no—
197C2×C4×D4—431no—
198C2×C4×D4C2×D4311yescharacteristic
199C22×C4⋊C4—432no—
200#431C2×D4312yes—
201#423—432no—
202C2×#100—861no—
203#343—861no—
204#332—864no—
205D4⋊C8—864no—
206#323—22no—
207C43#838711yescharacteristic, complemented, Sylow
208C2×#41—868no—
209C2×#41—1724no—
210C2×#41—1722no—
211C2×#41—1722no—
212C22×Q8—861no—
213C22×D4—431no—
214C22×D4—862no—
215C22×D4D17211yescharacteristic
216C23×C4—862no—
217C23×C4—861no—
218C23×C4C4×D4312yes—
219C2×M16—862no—
220C4⋊Q8—434no—
221C4⋊Q8—862no—
222#123—8616no—
223C22⋊Q8—868no—
224#121—864no—
225#121—864no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—434no—
229C4×D4—862no—
230C2×C4⋊C4—431no—
231C2×C4⋊C4—432no—
232C2×C4⋊C4—864no—
233C2×C4⋊C4—864no—
234C2×C4⋊C4—862no—
235C2×C4⋊C4—861no—
236C2×C4⋊C4D17211yescharacteristic
237C2×C22⋊C4—862no—
238C2×C22⋊C4—861no—
239C2×C22⋊C4C22⋊D4312yes—
240C2×C42C22×D4311yescharacteristic
241C4⋊C8—22no—
242C4⋊C8—862no—
243C4⋊C8—862no—
244#100—1722no—
245C22⋊C8—22no—
246C4×C8—22no—
247C24—861no—
248C24(C2×D43)⋊C412yes—
249#41—17216no—
250#41—17216no—
251C2×Q8—864no—
252C2×Q8—1721no—
253C2×Q8—1721no—
254C2×Q8—1722no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—432no—
258C2×D4—868no—
259C2×D4—861no—
260C2×D4—1722no—
261C2×D4—1724no—
262C2×D4—1724no—
263C2×D4#1062187312yes—
264C2×D4D34412yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—868no—
269C22×C4—861no—
270C22×C4—862no—
271C22×C4—864no—
272C22×C4—1724no—
273C22×C4—1724no—
274C22×C4—1721no—
275C22×C4—1721no—
276C22×C4—1722no—
277C22×C4(C2×D43)⋊C412yes—
278C22×C4C2×C4×D4311yescharacteristic
279C22×C4C2×D17211yescharacteristic
280C22×C4C2×C22⋊D4311yescharacteristic
281M16—1722no—
282M16—1722no—
283C2×C8—42no—
284C2×C8—1722no—
285C4⋊C4—22no—
286C4⋊C4—432no—
287C4⋊C4—434no—
288C4⋊C4—868no—
289C4⋊C4—861no—
290C4⋊C4—862no—
291C4⋊C4—864no—
292C22⋊C4—24no—
293C22⋊C4—868no—
294C22⋊C4—864no—
295C42D4×D4312yes—
296C42D4×D4312yes—
297C23—22no—
298C23—22no—
299C23—42no—
300C23—42no—
301C23—862no—
302C23—1722no—
303C23—1722no—
304C23—1721no—
305C23—1721no—
306C23#1306324714yes—
307C23C2×(C2×D43)⋊C411yescharacteristic
308Q8—1724no—
309Q8—1724no—
310D4—24no—
311D4—24no—
312D4—864no—
313D4—1722no—
314D4—17216no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—862no—
322C2×C4—864no—
323C2×C4—17216no—
324C2×C4—1721no—
325C2×C4—1722no—
326C2×C4—1721no—
327C2×C4—1722no—
328C2×C4—1724no—
329C2×C4—1724no—
330C2×C4C22⋊C4×D4311yescharacteristic
331C2×C4C22⋊C4×D4311yescharacteristic
332C2×C4#1306329312yes—
333C2×C4#1306329512yes—
334C2×C4C2×D34411yescharacteristic
335C2×C4C2×#1062187311yescharacteristic
336C2×C4#1306335312yes—
337C8—82no—
338C8—3442no—
339C22—21no—
340C22—21no—
341C22—24no—
342C22—44no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—3442no—
347C22—3442no—
348C22—1722no—
349C22—1722no—
350C22—1722no—
351C22#1458275411yescharacteristic
352C22C2×#1306324711yescharacteristic
353C22#1458290511yescharacteristic
354C4—21no—
355C4—22no—
356C4—42no—
357C4—1722no—
358C4—1724no—
359C4#1458288012yes—
360C4#1458303512yes—
361C4#1458305212yes—
362C2—21no—
363C2—21no—
364C2—44no—
365C2—3442no—
366C2#1913452011yescharacteristic
367C2#1913452711yescharacteristic
368C2#1913456811yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1460C2
socleC22×C43#379
prime radicalC2×#14583553C2
cosocleC2×C42×C43C23
hypercenter#1460D43
hypocenterC43#8387
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #19134587C2semidirect
3. #19133266C2semidirect
4. #19133021C2semidirect
5. C43×#1460C2semidirect
182. #1460D43semidirect
207. C43#8387semidirect

Maximal subgroups

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

subgroupindex
C2×#145835532
#191345872
#191332662
#191330212
C43×#14602
#838743