Finite Group Databasebeta

← #24993009

Subgroups — #24993009

Order 11008.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583099C211yesmaximal, characteristic
2#19135435C211yesmaximal, characteristic
3#19133264C211yesmaximal, characteristic
4#19133024C211yesmaximal, characteristic, complemented
5#19132469C211yesmaximal, characteristic, complemented
6#19132446C211yesmaximal, characteristic, complemented
7Q8×M16×C43C211yesmaximal, characteristic, complemented
8C22×#10621884C412yes—
9C2×#13063322C2211yescharacteristic
10#14583099—21no—
11#14583099—21no—
12#14583099—21no—
13#14583099—21no—
14#14582952C2211yescharacteristic
15#14582923C2211yescharacteristic
16C2×C4⋊(C43⋊C8)C2211yescharacteristic
17C2×#13063245C412yes—
18C2×#13063241—21no—
19C2×#13063241—21no—
20#14582642—22no—
21Q8⋊(C43⋊C8)—21no—
22Q8⋊(C43⋊C8)—21no—
23C2×C4×Q8×C43C2211yescharacteristic
24C43×#446—21no—
25C43×#446—22no—
26C8×Q8×C43—21no—
27C43×#423—21no—
28C43×#423C2211yescharacteristic
29C4×M16×C43—21no—
30C4×M16×C43C2211yescharacteristic
31C22×D172—21no—
32C22×D172—21no—
33C22×D172C2×C412yes—
34C2×#10621884—22no—
35C2×#10621884—22no—
36C2×#10621884—44no—
37C2×#10621877—21no—
38C22×C43⋊C8C2×C412yes—
39#13063322—21no—
40#13063322—21no—
41#13063322—21no—
42#13063322—21no—
43C4⋊(C43⋊C8)—21no—
44C4⋊(C43⋊C8)—21no—
45C4⋊(C43⋊C8)—21no—
46C4⋊(C43⋊C8)—21no—
47C4⋊(C43⋊C8)—21no—
48#13063245—22no—
49#13063245—22no—
50#13063241—42no—
51#13063241—42no—
52C22⋊(C43⋊C8)—21no—
53C4×C43⋊C8—21no—
54C22×Q8×C43C2×C411yescharacteristic
55C2×M16×C43—21no—
56C2×M16×C43D411yescharacteristic
57C2×M16×C43D411yescharacteristic
58C4×Q8×C43—21no—
59C4×Q8×C43—21no—
60C4×Q8×C43D411yescharacteristic
61C4×Q8×C43D411yescharacteristic
62C4×Q8×C43D411yescharacteristic
63C4×Q8×C43D411yescharacteristic
64C2×C4⋊C4×C43—21no—
65C2×C4⋊C4×C43C2×C411yescharacteristic
66C2×C42×C43—21no—
67C2×C42×C43C2311yescharacteristic
68C4⋊C8×C43—22no—
69C4⋊C8×C43—22no—
70C4⋊C8×C43—21no—
71C4⋊C8×C43D412yes—
72#97×C43—22no—
73#97×C43—21no—
74C4×C8×C43—22no—
75C4×C8×C43—21no—
76C23×D43—22no—
77C2×Q8×C43—21no—
78C2×Q8×C43—22no—
79C2×Q8×C43—41no—
80C2×Q8×C43C22⋊C411yescharacteristic
81C2×Q8×C43C22⋊C411yescharacteristic
82C22×C4×C43—21no—
83C22×C4×C43—21no—
84C22×C4×C43C22×C411yescharacteristic
85C22×C4×C43C2×D411yescharacteristic
86C22×C4×C43#4111yescharacteristic
87C2×D172—22no—
88C2×D172—22no—
89C2×D172—41no—
90C2×D172—41no—
91C2×D172—41no—
92C2×D172—41no—
93C2×D172—41no—
94C2×D172—41no—
95C2×D172—41no—
96C2×D172—41no—
97C2×D172—42no—
98C2×D172—42no—
99C2×D172—82no—
100M16×C43—21no—
101M16×C43—21no—
102M16×C43—21no—
103M16×C43—41no—
104M16×C43—41no—
105M16×C43SD1612yescomplemented
106C2×C8×C43—21no—
107C2×C8×C43—21no—
108C2×C8×C43—22no—
109C4⋊C4×C43—21no—
110C4⋊C4×C43—21no—
111C4⋊C4×C43—21no—
112C4⋊C4×C43—21no—
113C4⋊C4×C43—21no—
114C4⋊C4×C43C22⋊C411yescharacteristic
115C4⋊C4×C43C22⋊C411yescharacteristic
116C42×C43—21no—
117C42×C43—21no—
118C42×C43—21no—
119C42×C43—21no—
120C42×C43C2×D411yescharacteristic
121C42×C43C2×D411yescharacteristic
122C42×C43C2×D411yescharacteristic
123C42×C43#4111yescharacteristic
124#10621884—44no—
125#10621884—44no—
126#10621877—41no—
127#10621877—41no—
128C2×C43⋊C8—22no—
129C2×C43⋊C8—42no—
130C2×C43⋊C8—41no—
131C2×C43⋊C8—41no—
132C23×C43C4×D411yescharacteristic
133C22×D43—42no—
134C22×D43—42no—
135C22×D43—42no—
136C22×D43—82no—
137C22×D43—82no—
138Q8×C43—21no—
139Q8×C43—21no—
140Q8×C43—21no—
141Q8×C43—42no—
142Q8×C43#10012yescomplemented
143C2×C4×C43—21no—
144C2×C4×C43—21no—
145C2×C4×C43—21no—
146C2×C4×C43—21no—
147C2×C4×C43—21no—
148C2×C4×C43—21no—
149C2×C4×C43—41no—
150C2×C4×C43—42no—
151C2×C4×C43—81no—
152C2×C4×C43C2×C22⋊C411yescharacteristic
153C2×C4×C43C4×D411yescharacteristic
154C2×C4×C43#12011yescharacteristic
155C2×C4×C43#12111yescharacteristic
156C2×C4×C43#12111yescharacteristic
157C2×C4×C43#12311yescharacteristic
158C2×C4×C43C2×SD1611yescharacteristic
159C2×C4×C43#13511yescharacteristic
160C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C43#13711yescharacteristic
162D172—42no—
163D172—42no—
164D172—82no—
165D172—82no—
166D172—82no—
167D172—81no—
168D172—81no—
169D172—81no—
170D172—81no—
171C8×C43—22no—
172C8×C43—41no—
173C8×C43—42no—
174C43⋊C8—42no—
175C43⋊C8—81no—
176C43⋊C8—81no—
177#9254—431nomaximal, Sylow
178C22×C43—21no—
179C22×C43—21no—
180C22×C43#38611yescharacteristic
181C22×C43C4×SD1611yescharacteristic
182C22×C43#44011yescharacteristic
183C2×D43—82no—
184C2×D43—82no—
185C2×D43—82no—
186C2×D43—162no—
187C2×D43—162no—
188C4×C43—21no—
189C4×C43—22no—
190C4×C43—21no—
191C4×C43—21no—
192C4×C43—41no—
193C4×C43—41no—
194C4×C43—41no—
195C4×C43C2×#10011yescharacteristic
196C4×C43#41311yescharacteristic
197C4×C43#44811yescharacteristic
198C4×C43#44911yescharacteristic
199C4×C43#46011yescharacteristic
200C4×C43#46411yescharacteristic
201C2×#460—431no—
202Q8×M16D4311yescharacteristic, complemented
203#1965—431no—
204#1545—431no—
205#1487—431no—
206#1292—431no—
207#1283—431no—
208C2×C43—21no—
209C2×C43—21no—
210C2×C43#188011yescharacteristic
211C2×C43#190011yescharacteristic
212C2×C43#190311yescharacteristic
213D43—162no—
214C22×SD16—432no—
215C2×#127—431no—
216C2×C4×Q8C2×D4311yescharacteristic
217#460—861no—
218#460—861no—
219#460—861no—
220#460—861no—
221#446—21no—
222#446—22no—
223C8×Q8—21no—
224#431—431no—
225#423—21no—
226#423—431no—
227#423C2×D4311yescharacteristic
228C2×C4⋊C8—431no—
229C2×#102—432no—
230C2×#100—861no—
231C2×#100—861no—
232C4×M16—21no—
233C4×M16C2×D4311yescharacteristic
234#331—862no—
235Q8⋊C8—861no—
236Q8⋊C8—861no—
237C43#925411yescharacteristic, complemented, Sylow
238C22×Q8C4×D4311yescharacteristic
239C22×D4—432no—
240C22×D4—861no—
241C22×D4—861no—
242C2×SD16—862no—
243C2×SD16—862no—
244C2×SD16—1724no—
245C2×M16—21no—
246C2×M16—861no—
247C2×M16C22⋊D4311yescharacteristic
248C2×M16C22⋊D4311yescharacteristic
249C22×C8—432no—
250#127—861no—
251#127—861no—
252#127—861no—
253#127—861no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D17211yescharacteristic
257C4×Q8D17211yescharacteristic
258C4×Q8C22⋊D4311yescharacteristic
259C4×Q8C22⋊D4311yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D4311yescharacteristic
262C2×C42—21no—
263C2×C42C22×D4311yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—861no—
268C4⋊C8—861no—
269C4⋊C8—861no—
270C4⋊C8—861no—
271C4⋊C8—861no—
272C4⋊C8C22⋊D4312yes—
273#102—862no—
274#102—862no—
275#100—1722no—
276#100—1722no—
277C22⋊C8—861no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—861no—
283C24—862no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8(C2×D43)⋊C411yescharacteristic
288C2×Q8(C2×D43)⋊C411yescharacteristic
289C2×D4—862no—
290C2×D4—862no—
291C2×D4—3442no—
292C2×D4—1722no—
293C2×D4—1722no—
294C2×D4—1721no—
295C2×D4—1721no—
296C2×D4—1721no—
297C2×D4—1721no—
298C2×D4—1721no—
299C2×D4—1721no—
300C2×D4—1721no—
301C2×D4—1721no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D4311yescharacteristic
305C22×C4#1062189811yescharacteristic
306C22×C4C2×C22⋊D4311yescharacteristic
307SD16—1724no—
308SD16—1724no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—1721no—
315M16—1721no—
316M16#1062188412yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—862no—
321C2×C8—1721no—
322C2×C8—1721no—
323C2×C8—1722no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4(C2×D43)⋊C411yescharacteristic
330C4⋊C4(C2×D43)⋊C411yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D17211yescharacteristic
336C42#1062189811yescharacteristic
337C42C2×C22⋊D4311yescharacteristic
338C42C2×C22⋊D4311yescharacteristic
339C23—3442no—
340C23—3442no—
341C23—1722no—
342C23—1722no—
343C23—1722no—
344C23C4×C22⋊D4311yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1306324112yescomplemented
350D4—3442no—
351D4—3442no—
352D4—3442no—
353D4—3441no—
354D4—3441no—
355D4—3441no—
356D4—3441no—
357D4—1722no—
358D4—1722no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×(C2×D43)⋊C411yescharacteristic
369C2×C4C4×C22⋊D4311yescharacteristic
370C2×C4#1306329411yescharacteristic
371C2×C4#1306329911yescharacteristic
372C2×C4#1306329911yescharacteristic
373C2×C4#1306330811yescharacteristic
374C2×C4C2×#1062188411yescharacteristic
375C2×C4#1306334511yescharacteristic
376C2×C4#1306335011yescharacteristic
377C2×C4#1306335611yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—3441no—
382C8—3441no—
383C8—1722no—
384C22—21no—
385C22—21no—
386C22—3442no—
387C22—3442no—
388C22—3442no—
389C22—6882no—
390C22—6882no—
391C22#1458277811yescharacteristic
392C22C4×#1062188411yescharacteristic
393C22#1458299711yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1306324111yescharacteristic
402C4#1458288111yescharacteristic
403C4#1458303811yescharacteristic
404C4#1458304411yescharacteristic
405C4#1458309911yescharacteristic
406C4#1458312111yescharacteristic
407C2—21no—
408C2—21no—
409C2—6882no—
410C2#1913495011yescharacteristic
411C2#1913505811yescharacteristic
412C2#1913507511yescharacteristic
subgroupG/H
centerC22#14582778
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingQ8×M16×C43C2
socleC22×C43#386
prime radicalC2×#13063322C22
cosocleC2×C42×C43C23
hypercenterQ8×M16D43
hypocenterC43#9254
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #19133024C2semidirect
5. #19132469C2semidirect
6. #19132446C2semidirect
7. Q8×M16×C43C2semidirect
105. M16×C43SD16semidirect
142. Q8×C43#100semidirect
202. Q8×M16D43semidirect
237. C43#9254semidirect
316. M16#10621884semidirect
349. Q8#13063241semidirect

Maximal subgroups

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

subgroupindex
C2×#145830992
#191354352
#191332642
#191330242
#191324692
#191324462
Q8×M16×C432
#925443