Finite Group Databasebeta

← #24993751

Subgroups — #24993751

Order 11008.

538 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144075C211yesmaximal, characteristic
2#19135434C211yesmaximal, characteristic, complemented
3#19133262C211yesmaximal, characteristic, complemented
4#19133034C211yesmaximal, characteristic, complemented
5C43⋊#1309C211yesmaximal, characteristic, complemented
6#19132486C211yesmaximal, characteristic, complemented
7C43×#2991C211yesmaximal, characteristic, complemented
8C2×#13063350C412yes—
9C2×#13063322C2211yescharacteristic
10#14583099—21no—
11#14583099—21no—
12#14583094—21no—
13#14583094—21no—
14#14582923C2211yescharacteristic
15#14582923C2211yescharacteristic
16#14582903C412yes—
17C2×#13063241—22no—
18C4×#10621877C2211yescharacteristic
19#14582642—21no—
20#14582642—21no—
21Q8⋊(C43⋊C8)—21no—
22D4⋊(C43⋊C8)—21no—
23C4×#41×C43C2211yescharacteristic
24C8×Q8×C43—21no—
25C43×#436—22no—
26C43×#436—21no—
27C43×#431—21no—
28C43×#431C2211yescharacteristic
29C43×#423—21no—
30C43×#423C2211yescharacteristic
31C22×D172—21no—
32C22×D172—21no—
33C22×D172C2×C412yes—
34#13063350—44no—
35C2×#10621884—22no—
36C2×#10621882—22no—
37C2×#10621877—21no—
38C2×#10621877—21no—
39C2×#10621877C2×C412yes—
40#13063322—21no—
41#13063322—21no—
42#13063322—21no—
43#13063322—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—44no—
51#13063236—21no—
52C4×C43⋊C8—21no—
53C2×#41×C43C2×C411yescharacteristic
54C2×M16×C43—22no—
55C2×M16×C43D412yes—
56C4×Q8×C43D411yescharacteristic
57C4×Q8×C43D411yescharacteristic
58C4×D4×C43—21no—
59C4×D4×C43—21no—
60C4×D4×C43D411yescharacteristic
61C4×D4×C43D411yescharacteristic
62#117×C43—21no—
63#117×C43C2×C411yescharacteristic
64C2×C42×C43—21no—
65C2×C42×C43C2311yescharacteristic
66C4⋊C8×C43—22no—
67C4⋊C8×C43—21no—
68C4⋊C8×C43—21no—
69C4⋊C8×C43D411yescharacteristic
70C4⋊C8×C43D411yescharacteristic
71C22⋊C8×C43—22no—
72C22⋊C8×C43—21no—
73C4×C8×C43—22no—
74C4×C8×C43—21no—
75C23×D43—22no—
76#41×C43—22no—
77#41×C43—41no—
78C2×Q8×C43C22⋊C411yescharacteristic
79C2×D4×C43—21no—
80C2×D4×C43C22⋊C411yescharacteristic
81C22×C4×C43—21no—
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—42no—
90C2×D172—42no—
91C2×D172—41no—
92C2×D172—41no—
93C2×D172—41no—
94C2×D172—41no—
95C2×D172—42no—
96C2×D172—42no—
97C2×D172—81no—
98C2×D172—81no—
99M16×C43—42no—
100M16×C43—82no—
101C2×C8×C43—21no—
102C2×C8×C43—22no—
103C2×C8×C43—21no—
104C4⋊C4×C43—21no—
105C4⋊C4×C43—21no—
106C4⋊C4×C43C22⋊C411yescharacteristic
107C4⋊C4×C43C22⋊C411yescharacteristic
108C22⋊C4×C43—21no—
109C22⋊C4×C43—21no—
110C22⋊C4×C43—21no—
111C42×C43—21no—
112C42×C43—21no—
113C42×C43—21no—
114C42×C43C2×D411yescharacteristic
115C42×C43C2×D411yescharacteristic
116C42×C43C2×D411yescharacteristic
117C42×C43#4111yescharacteristic
118#10621884—44no—
119#10621882—44no—
120#10621877—22no—
121#10621877—41no—
122#10621877—42no—
123#10621877—81no—
124C2×C43⋊C8—22no—
125C2×C43⋊C8—41no—
126C2×C43⋊C8—41no—
127C23×C43—21no—
128C23×C43C4×D411yescharacteristic
129C22×D43—42no—
130C22×D43—44no—
131C22×D43—82no—
132C22×D43—82no—
133Q8×C43—21no—
134Q8×C43#10012yescomplemented
135D4×C43—21no—
136D4×C43—21no—
137D4×C43—42no—
138C2×C4×C43—21no—
139C2×C4×C43—21no—
140C2×C4×C43—21no—
141C2×C4×C43—21no—
142C2×C4×C43—21no—
143C2×C4×C43—21no—
144C2×C4×C43—21no—
145C2×C4×C43—21no—
146C2×C4×C43—21no—
147C2×C4×C43—42no—
148C2×C4×C43—41no—
149C2×C4×C43—41no—
150C2×C4×C43—41no—
151C2×C4×C43—41no—
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
158D172—42no—
159D172—42no—
160D172—82no—
161D172—82no—
162D172—82no—
163D172—81no—
164D172—81no—
165D172—81no—
166D172—81no—
167C8×C43—42no—
168C8×C43—41no—
169C8×C43—41no—
170C43⋊C8—42no—
171C43⋊C8—42no—
172C43⋊C8—81no—
173#9360—431nomaximal, Sylow
174C22×C43—21no—
175C22×C43—21no—
176C22×C43—21no—
177C22×C43—41no—
178C22×C43—41no—
179C22×C43—41no—
180C22×C43#38611yescharacteristic
181C2×D43—82no—
182C2×D43—82no—
183C2×D43—82no—
184C2×D43—164no—
185C4×C43—21no—
186C4×C43—21no—
187C4×C43—22no—
188C4×C43—21no—
189C4×C43—41no—
190C4×C43—41no—
191C4×C43C2×#10011yescharacteristic
192C4×C43#41311yescharacteristic
193C4×C43#45311yescharacteristic
194C4×C43#45411yescharacteristic
195C4×C43#47111yescharacteristic
196C4×C43#47211yescharacteristic
197#3038—431no—
198#2991D4311yescharacteristic, complemented
199#1965—431no—
200#1545—431no—
201#1489—431no—
202#1309—431no—
203#1299—431no—
204C2×C43—41no—
205C2×C43—81no—
206C2×C43#188311yescharacteristic
207C2×C43#190011yescharacteristic
208C2×C43#190711yescharacteristic
209D43—162no—
210C2×AGL(1,ℤ/8)—432no—
211C2×#127—431no—
212C4×#41C2×D4311yescharacteristic
213#460—861no—
214#460—861no—
215#459—861no—
216#459—861no—
217C8×Q8—21no—
218#436—22no—
219#436—21no—
220#431—21no—
221#431C2×D4311yescharacteristic
222#423—21no—
223#423—431no—
224#423—431no—
225#423C2×D4311yescharacteristic
226#418—432no—
227C2×#100—862no—
228C4×M16—431no—
229#331—861no—
230#331—861no—
231Q8⋊C8—861no—
232D4⋊C8—861no—
233C43#936011yescharacteristic, complemented, Sylow
234C2×#41C4×D4311yescharacteristic
235C22×D4—432no—
236C22×D4—861no—
237C22×D4—861no—
238AGL(1,ℤ/8)—1724no—
239C2×SD16—862no—
240C2×D8—862no—
241C2×M16—22no—
242C2×M16—432no—
243C2×M16—861no—
244C2×M16—861no—
245C2×M16C22⋊D4312yes—
246#127—861no—
247#127—861no—
248#127—861no—
249#127—861no—
250C4×Q8D17211yescharacteristic
251C4×Q8C22⋊D4311yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4D17211yescharacteristic
255C4×D4C22⋊D4311yescharacteristic
256#117—21no—
257#117C4×D4311yescharacteristic
258C2×C42—21no—
259C2×C42C22×D4311yescharacteristic
260C4⋊C8—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—861no—
264C4⋊C8—861no—
265C4⋊C8—861no—
266C4⋊C8—861no—
267C4⋊C8C22⋊D4311yescharacteristic
268C4⋊C8C22⋊D4311yescharacteristic
269#102—862no—
270#102—862no—
271#100—1724no—
272C22⋊C8—22no—
273C22⋊C8—21no—
274#97—861no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—861no—
278C24—862no—
279#41—22no—
280#41—41no—
281C2×Q8(C2×D43)⋊C411yescharacteristic
282C2×D4—21no—
283C2×D4—862no—
284C2×D4—862no—
285C2×D4—3441no—
286C2×D4—3441no—
287C2×D4—1722no—
288C2×D4—1722no—
289C2×D4—1722no—
290C2×D4—1722no—
291C2×D4—1721no—
292C2×D4—1721no—
293C2×D4—1721no—
294C2×D4—1721no—
295C2×D4(C2×D43)⋊C411yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C2×C4×D4311yescharacteristic
300C22×C4#1062189811yescharacteristic
301C22×C4C2×C22⋊D4311yescharacteristic
302SD16—1724no—
303D8—1724no—
304M16—42no—
305M16—82no—
306M16—862no—
307M16—3441no—
308M16—1721no—
309M16—1722no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—862no—
314C2×C8—1721no—
315C2×C8—1721no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4(C2×D43)⋊C411yescharacteristic
319C4⋊C4(C2×D43)⋊C411yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C2×D17211yescharacteristic
327C42#1062189811yescharacteristic
328C42C2×C22⋊D4311yescharacteristic
329C42C2×C22⋊D4311yescharacteristic
330C23—21no—
331C23—3442no—
332C23—3442no—
333C23—1724no—
334C23—1722no—
335C23C4×C22⋊D4311yescharacteristic
336Q8—21no—
337Q8#1306324112yescomplemented
338D4—21no—
339D4—21no—
340D4—42no—
341D4—3442no—
342D4—3442no—
343D4—3442no—
344D4—3441no—
345D4—3441no—
346D4—3441no—
347D4—3441no—
348D4—1722no—
349D4—1722no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C2×(C2×D43)⋊C411yescharacteristic
365C2×C4C4×C22⋊D4311yescharacteristic
366C2×C4#1306329411yescharacteristic
367C2×C4#1306329911yescharacteristic
368C2×C4#1306329911yescharacteristic
369C2×C4#1306330811yescharacteristic
370C8—42no—
371C8—41no—
372C8—41no—
373C8—3441no—
374C8—1722no—
375C8—1722no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—3442no—
383C22—3442no—
384C22—3442no—
385C22—6884no—
386C22#1458277811yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4C2×#1306324111yescharacteristic
394C4#1458288111yescharacteristic
395C4#1458306711yescharacteristic
396C4#1458307211yescharacteristic
397C4#1458316211yescharacteristic
398C4#1458316911yescharacteristic
399C2—41no—
400C2—81no—
401C2—6882no—
402C2#1913497111yescharacteristic
403C2#1913505811yescharacteristic
404C2#1913510311yescharacteristic
subgroupG/H
centerC22#14582778
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#2991C2
socleC22×C43#386
prime radical#19144075C2
cosocleC2×C42×C43C23
hypercenter#2991D43
hypocenterC43#9360
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #19135434C2semidirect
3. #19133262C2semidirect
4. #19133034C2semidirect
5. C43⋊#1309C2semidirect
6. #19132486C2semidirect
7. C43×#2991C2semidirect
134. Q8×C43#100semidirect
198. #2991D43semidirect
233. C43#9360semidirect
337. Q8#13063241semidirect

Maximal subgroups

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

subgroupindex
#191440752
#191354342
#191332622
#191330342
C43⋊#13092
#191324862
C43×#29912
#936043