Finite Group Databasebeta

← #37547489

Subgroups — #37547489

Order 23168.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153864C211yesmaximal, characteristic, complemented
2C2×#19153863C211yesmaximal, characteristic, complemented
3#26041948C211yesmaximal, characteristic, complemented
4#26041947C213yesmaximal, complemented
5Q8×C22⋊D181C211yesmaximal, characteristic, complemented
6#26041942C213yesmaximal, complemented
7#26041941C211yesmaximal, characteristic, complemented
8#26041938C213yesmaximal, complemented
9#26041937C213yesmaximal, complemented
10#26041859C213yesmaximal, complemented
11#26041858C213yesmaximal, complemented
12#26041828C213yesmaximal, complemented
13#26041826C213yesmaximal, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16#19153864—28no—
17#19153863—28no—
18C2×#15668450C2213yes—
19C2×#15668450C2211yescharacteristic
20C2×#15668449C2213yes—
21C2×#15668449C2213yes—
22C2×#15668449C2213yes—
23C2×#15668448C2213yes—
24C2×#15668448C2213yes—
25C22×Dic362C2213yes—
26C2×Q8×D181C2211yescharacteristic
27C2×Q8×D181C2213yes—
28C2×Q8×D181C2211yescharacteristic
29#19153815C2213yes—
30#19153814C2213yes—
31#19153812C2213yes—
32#19153812C2213yes—
33#19153804C2213yes—
34#19153802C2213yes—
35#19153802C2213yes—
36#19153796C2213yes—
37#19153795C2213yes—
38#19153794C2216yes—
39#19153793C2216yes—
40#19153792C2213yes—
41#19153791C2213yes—
42#19153791C2213yes—
43#19153790C2213yes—
44#19153789C2216yes—
45#19153789C2216yes—
46#19153789C2211yescharacteristic
47#19153789C2213yes—
48#19153788C2216yes—
49#19153787C2213yes—
50#19153787C2213yes—
51Q8×Dic181C2211yescharacteristic
52C4×Dic362C2213yes—
53#19153781C2213yes—
54C4×D724C2213yes—
55C4×C22⋊D181C2213yes—
56C4×C22⋊D181C2213yes—
57#19153776C2216yes—
58#19153776C2216yes—
59C4⋊C4×D181C2213yes—
60C42×D181C2213yes—
61C2×#41×C181C2211yescharacteristic
62C22×Q8×C181C2211yescharacteristic
63#124×C181C2213yes—
64C22⋊Q8×C181C2213yes—
65#121×C181C2213yes—
66C4×Q8×C181C2211yescharacteristic
67C4×D4×C181C2213yes—
68#15668450—212no—
69#15668450—24no—
70#15668449—212no—
71#15668449—212no—
72#15668449—212no—
73#15668448—212no—
74#15668448—212no—
75C2×Dic362—212no—
76C2×Dic362C2313yes—
77C2×Dic362C2313yes—
78C2×Dic362C2313yes—
79C2×Dic362C2313yes—
80Q8×D181—212no—
81Q8×D181—24no—
82Q8×D181D418yescomplemented
83C2×D724C2313yes—
84C2×D724C2311yescharacteristic
85C2×D724C2313yes—
86C2×D724C2313yes—
87C2×C22⋊D181C2313yes—
88C2×C22⋊D181C2311yescharacteristic
89C2×C22⋊D181C2311yescharacteristic
90C2×C22⋊D181C2313yes—
91C2×C22⋊D181C2313yes—
92C2×C22⋊D181C2313yes—
93C2×C22⋊D181C2316yes—
94C22×Dic181C2316yes—
95C2×C4×D181C2311yescharacteristic
96C2×C4×D181C2316yes—
97C2×C4×D181C2313yes—
98C2×C4×D181C2313yes—
99C2×C4×D181C2313yes—
100C2×C4×D181C2313yes—
101C4⋊Dic181C2313yes—
102C4⋊Dic181C2313yes—
103#15668428C2316yes—
104#15668428C2313yes—
105#15668428C2316yes—
106#15668428C2311yescharacteristic
107C22⋊Dic181C2313yes—
108C22⋊Dic181C2313yes—
109#15668426C2313yes—
110#15668426C2316yes—
111#15668426C2313yes—
112#15668426C2313yes—
113#15668426C2311yescharacteristic
114C4×Dic181C2313yes—
115C4×Dic181C2313yes—
116C4×Dic181C2313yes—
117#41×C181—24no—
118C2×Q8×C181—24no—
119C2×Q8×C181C2313yes—
120C2×Q8×C181C2311yescharacteristic
121C2×D4×C181C2313yes—
122C2×D4×C181C2313yes—
123C22×C4×C181C2313yes—
124C22×C4×C181C2313yes—
125C4⋊C4×C181C2313yes—
126C4⋊C4×C181C2313yes—
127C22⋊C4×C181C2311yescharacteristic
128C22⋊C4×C181C2313yes—
129C22⋊C4×C181C2316yes—
130C42×C181C2313yes—
131C22×D181C2411yescharacteristic
132C22×D181C2413yes—
133C22×D181C2411yescharacteristic
134Dic362—26no—
135Dic362—26no—
136Dic362—26no—
137Dic362C2×D4112yes—
138D724—22no—
139D724—26no—
140D724—26no—
141D724—43no—
142C22⋊D181—26no—
143C22⋊D181—22no—
144C22⋊D181—26no—
145C22⋊D181—26no—
146C22⋊D181—26no—
147C22⋊D181—22no—
148C22⋊D181—46no—
149C2×Dic181—212no—
150C2×Dic181C2413yes—
151C2×Dic181C2413yes—
152C2×Dic181C2411yescharacteristic
153C2×Dic181C2413yes—
154C2×Dic181C2411yescharacteristic
155C4×D181—26no—
156C4×D181—26no—
157C4×D181—212no—
158C4×D181—22no—
159C4×D181—43no—
160C4×D181C2×D4112yes—
161C23×C181C2413yes—
162C23×C181C2411yescharacteristic
163Q8×C181—26no—
164Q8×C181C2×D414yes—
165D4×C181—26no—
166D4×C181—43no—
167C2×C4×C181—26no—
168C2×C4×C181—26no—
169C2×C4×C181C2411yescharacteristic
170C2×C4×C181C2413yes—
171C2×C4×C181C2413yes—
172C2×C4×C181C2411yescharacteristic
173C2×C4×C181C2413yes—
174C22×C181—21no—
175C22×C181—23no—
176C22×C181—23no—
177C22×C181—21no—
178C22×C181—21no—
179C22×C181—23no—
180C22×C181C2511yescharacteristic
181C2×D181—21no—
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—23no—
185C2×D181—21no—
186C2×D181—23no—
187C2×D181—23no—
188C2×D181—21no—
189C2×D181C22×D412yes—
190C4×C181—21no—
191C4×C181—23no—
192C4×C181—21no—
193C4×C181—23no—
194C4×C181C22×D416yes—
195Dic181—21no—
196Dic181—23no—
197Dic181—23no—
198Dic181C22×D416yes—
199Dic1812−1+412yescomplemented
200C2×C181—41no—
201C2×C181—43no—
202C2×C181C23×D411yescharacteristic
203C2×C181C2×2−1+411yescharacteristic
204C2×C181#58511yescharacteristic
205D181—22no—
206D181—41no—
207D181—43no—
208C181#349711yescharacteristic, complemented, Sylow
209#3497—1811nomaximal, Sylow
210C2×2−1+4—1811no—
211C2×2−1+4—1811no—
212#550—1811no—
213#550—1813no—
214D4×Q8—1811no—
215D4×Q8—1813no—
216#548—1813no—
217#548—1811no—
218#548—1813no—
219#548D18111yescharacteristic, complemented
220#537—1813no—
221#537—1813no—
222#532—1813no—
223#532—1813no—
224C2×C4×Q8—1811no—
2252−1+4—3628no—
2262−1+4—3628no—
227C2×#41—1813no—
228C2×#41—1811no—
229C2×#41—1813no—
230C2×#41—1813no—
231C2×#41—1813no—
232C2×#41—1813no—
233C2×#41—1813no—
234C2×#41C2×D18111yescharacteristic
235C22×Q8—1813no—
236C22×Q8—1811no—
237C22×Q8—1811no—
238C22×Q8—1813no—
239C22×Q8C2×D18111yescharacteristic
240C4⋊Q8—1813no—
241C4⋊Q8—1813no—
242#127—1813no—
243#127—1813no—
244#124—1813no—
245#124—1813no—
246#124—1813no—
247#124C2×D18113yes—
248C22⋊Q8—1813no—
249C22⋊Q8—1813no—
250C22⋊Q8—1816no—
251C22⋊Q8—1813no—
252C22⋊Q8—1816no—
253C22⋊Q8C2×D18113yes—
254#121—1813no—
255#121—1813no—
256#121—1816no—
257#121—1813no—
258#121—1813no—
259#121—1813no—
260#121—1813no—
261#121—1811no—
262#121—1816no—
263#121—1816no—
264#121C2×D18113yes—
265C4×Q8—1811no—
266C4×Q8—1813no—
267C4×Q8—1813no—
268C4×Q8C2×D18111yescharacteristic
269C4×D4—1813no—
270C4×D4—1813no—
271C4×D4—1816no—
272C4×D4—1816no—
273C4×D4—1813no—
274C4×D4C2×D18113yes—
275C2×C4⋊C4—1813no—
276C2×C42—1813no—
277#41—24no—
278#41—36212no—
279#41—36212no—
280#41—36212no—
281#41—36212no—
282#41—3624no—
283#41—36212no—
284#41—36212no—
285C2×Q8—24no—
286C2×Q8—36212no—
287C2×Q8—36212no—
288C2×Q8—3624no—
289C2×Q8—1818no—
290C2×Q8—1813no—
291C2×Q8—1813no—
292C2×Q8—1813no—
293C2×Q8—1813no—
294C2×Q8C22×D18113yes—
295C2×Q8C22×D18111yescharacteristic
296C2×D4—1811no—
297C2×D4—1813no—
298C2×D4—1811no—
299C2×D4—1813no—
300C2×D4—1813no—
301C2×D4—1813no—
302C2×D4—1816no—
303C2×D4—1811no—
304C2×D4—1813no—
305C2×D4—1813no—
306C2×D4—1813no—
307C2×D4C22×D18113yes—
308C2×D4C22×D18113yes—
309C22×C4—1816no—
310C22×C4—1811no—
311C22×C4—1816no—
312C22×C4—1813no—
313C22×C4—1813no—
314C22×C4—1813no—
315C22×C4—1813no—
316C22×C4C22×D18113yes—
317C22×C4C22×D18113yes—
318C4⋊C4—1816no—
319C4⋊C4—1813no—
320C4⋊C4—1811no—
321C4⋊C4—1813no—
322C4⋊C4—1816no—
323C4⋊C4—1813no—
324C4⋊C4C22×D18113yes—
325C4⋊C4C22×D18113yes—
326C22⋊C4—1813no—
327C22⋊C4—1813no—
328C22⋊C4—1813no—
329C22⋊C4—1811no—
330C22⋊C4—1813no—
331C22⋊C4—1813no—
332C22⋊C4—1816no—
333C22⋊C4C22×D18113yes—
334C22⋊C4C22×D18116yes—
335C22⋊C4C22×D18111yescharacteristic
336C42—1813no—
337C42—1813no—
338C42—1813no—
339C42C22×D18113yes—
340C23—1811no—
341C23—1813no—
342C23—1811no—
343C23C23×D18111yescharacteristic
344C23C23×D18113yes—
345Q8—26no—
346Q8—3626no—
347Q8—3626no—
348Q8—3626no—
349Q8—18112no—
350Q8D4×D18114yes—
351D4—26no—
352D4—43no—
353D4—3622no—
354D4—3626no—
355D4—3622no—
356D4—3626no—
357D4—3626no—
358D4—3626no—
359D4—3622no—
360D4—3626no—
361D4—3626no—
362D4—7243no—
363D4—7246no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—36212no—
367C2×C4—3622no—
368C2×C4—36212no—
369C2×C4—3626no—
370C2×C4—3626no—
371C2×C4—1813no—
372C2×C4—1811no—
373C2×C4—1813no—
374C2×C4—1811no—
375C2×C4—1813no—
376C2×C4—18112no—
377C2×C4—7243no—
378C2×C4C23×D18111yescharacteristic
379C2×C4C23×D18113yes—
380C2×C4C23×D18111yescharacteristic
381C2×C4C23×D18113yes—
382C2×C4C23×D18113yes—
383C22—21no—
384C22—23no—
385C22—21no—
386C22—23no—
387C22—21no—
388C22—23no—
389C22—3621no—
390C22—3621no—
391C22—3621no—
392C22—3623no—
393C22—3623no—
394C22—3621no—
395C22—3621no—
396C22—3623no—
397C22—1812no—
398C22C24×D18111yescharacteristic
399C4—23no—
400C4—21no—
401C4—23no—
402C4—21no—
403C4—3623no—
404C4—3623no—
405C4—3621no—
406C4—1816no—
407C4—1812no—
408C4C2×D4×D18116yes—
409C2—41no—
410C2—43no—
411C2—3622no—
412C2—7241no—
413C2—7243no—
414C2C22×D4×D18111yescharacteristic
415C22−1+4×D18111yescharacteristic
416C2#2604211511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#548×C181C2
hypercenter#548D181
hypocenterC181#3497
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19153864C2semidirect
2. C2×#19153863C2semidirect
3. #26041948C2semidirect
4. #26041947C2semidirect
5. Q8×C22⋊D181C2semidirect
6. #26041942C2semidirect
7. #26041941C2semidirect
8. #26041938C2semidirect
9. #26041937C2semidirect
10. #26041859C2semidirect
11. #26041858C2semidirect
12. #26041828C2semidirect
13. #26041826C2semidirect
14. C4×Q8×D181C2semidirect
15. #548×C181C2semidirect
82. Q8×D181D4semidirect
199. Dic1812−1+4semidirect
208. C181#3497semidirect
219. #548D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
C2×#191538632
#260419482
#260419472
Q8×C22⋊D1812
#260419422
#260419412
#260419382
#260419372
#260418592
#260418582
#260418282
#260418262
C4×Q8×D1812
#548×C1812
#3497181