Finite Group Databasebeta

← #37545274

Subgroups — #37545274

Order 23168.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153846C211yesmaximal, characteristic, complemented
2C2×#19153835C211yesmaximal, characteristic, complemented
3#26041935C211yesmaximal, characteristic, complemented
4#26041928C211yesmaximal, characteristic, complemented
5#26041670C211yesmaximal, characteristic, complemented
6#26041523C211yesmaximal, characteristic, complemented
7#26041519C211yesmaximal, characteristic, complemented
8#26041492C211yesmaximal, characteristic, complemented
9#26041488C211yesmaximal, characteristic, complemented
10#26041442C211yesmaximal, characteristic, complemented
11#26041438C211yesmaximal, characteristic, complemented
12#26041437C211yesmaximal, characteristic, complemented
13#26041376C211yesmaximal, characteristic, complemented
14#26041353C211yesmaximal, characteristic, complemented
15#449×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C22×Dic362C2211yescharacteristic
19#19153846—24no—
20#19153835—24no—
21C2×Dic724C2211yescharacteristic
22C2×Dic724C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668437C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668433C2211yescharacteristic
29C2×C8×D181C2211yescharacteristic
30#19153814C2211yescharacteristic
31#19153803C2211yescharacteristic
32#19153802—21no—
33#19153799—21no—
34#19153797—21no—
35#19153796C2211yescharacteristic
36#19153795C2211yescharacteristic
37#19153793—21no—
38#19153792C2211yescharacteristic
39#19153788—21no—
40#19153788—21no—
41#19153781C2211yescharacteristic
42D4×Dic181—21no—
43D4×Dic181C2211yescharacteristic
44C4×C22⋊D181C2211yescharacteristic
45C2×#15668428—21no—
46#19153749C2211yescharacteristic
47#19153745C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153741C2211yescharacteristic
50#19153739C2211yescharacteristic
51#19153737C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153726C2211yescharacteristic
54C2×#41×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C2×D8×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#102×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic
61#15668448—22no—
62#15668448—22no—
63#15668448—41no—
64#15668448—41no—
65C2×Dic362—22no—
66C2×Dic362—21no—
67C2×Dic362—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×Dic362C2311yescharacteristic
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181D411yescharacteristic
74C22×Dic181—21no—
75C22×Dic181—21no—
76C22×Dic181—21no—
77C22×Dic181—21no—
78C2×C4×D181C2311yescharacteristic
79Dic724—22no—
80Dic724—41no—
81#15668439—22no—
82#15668439—22no—
83#15668437—22no—
84#15668437—22no—
85#15668437D414yescomplemented
86#15668433—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C8×D181—22no—
90C4⋊Dic181—21no—
91C4⋊Dic181C2311yescharacteristic
92#15668428—21no—
93#15668428—21no—
94#15668428—21no—
95#15668428—21no—
96#15668428—21no—
97#15668428D411yescharacteristic
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181D411yescharacteristic
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108#41×C181—22no—
109#41×C181—41no—
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—41no—
118D8×C181—22no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181C2×D411yescharacteristic
124Dic362—21no—
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—21no—
129Dic362—42no—
130Dic362C2×D412yes—
131C22⋊D181—41no—
132C22⋊D181—42no—
133C22⋊D181—42no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—42no—
139C2×Dic181—42no—
140C2×Dic181—42no—
141C2×Dic181—42no—
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C4×D181—21no—
146C4×D181—21no—
147#13089678—21no—
148#13089678C2×D412yes—
149C23×C181—21no—
150C23×C181—21no—
151C23×C181C2×D411yescharacteristic
152Q8×C181—21no—
153Q8×C181—21no—
154D4×C181—21no—
155D4×C181—21no—
156D4×C181—21no—
157D4×C181—42no—
158D4×C181—42no—
159D4×C181C2×D412yes—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—42no—
165C2×C4×C181C2×D411yescharacteristic
166C2×C4×C181C2411yescharacteristic
167C8×C181—21no—
168C8×C181—21no—
169C22×C181—21no—
170C22×C181—22no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181C22×D411yescharacteristic
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—21no—
182C4×C181—21no—
183C4×C181—41no—
184C4×C181—41no—
185C4×C181C22×D411yescharacteristic
186C4×C181H(5,𝔽2)11yescharacteristic
187Dic181—21no—
188Dic181—22no—
189Dic181—21no—
190Dic181—21no—
191Dic181—41no—
192Dic181—41no—
193Dic181—41no—
194C2×C181—41no—
195C2×C181—42no—
196C2×C181—81no—
197C2×C181D4211yescharacteristic
198C2×C181#57511yescharacteristic
199C2×C181#57811yescharacteristic
200D181—41no—
201C181#326911yescharacteristic, complemented, Sylow
202#3269—1811nomaximal, Sylow
203C2×#137—1811no—
204C2×#135—1811no—
205#548—1811no—
206#547—1811no—
207#497—1811no—
208#468—1811no—
209#467—1811no—
210#463—1811no—
211#462—1811no—
212#452—1811no—
213#452—1811no—
214#451—1811no—
215#449D18111yescharacteristic, complemented
216#440—1811no—
217#435—1811no—
218C2×#41—1811no—
219C2×#41—1811no—
220C2×#41C2×D18111yescharacteristic
221C22×Q8—1811no—
222#137—3624no—
223#135—3624no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×SD16—1811no—
229C2×SD16—1811no—
230C2×SD16—1811no—
231C2×SD16C2×D18111yescharacteristic
232C2×D8C2×D18111yescharacteristic
233C2×M16—1811no—
234C22×C8—1811no—
235C4⋊Q8—1811no—
236#124—3621no—
237#124—1811no—
238#123—3621no—
239#123—3621no—
240C22⋊Q8—3621no—
241C22⋊Q8—1811no—
242C22⋊Q8—1811no—
243C22⋊Q8—1811no—
244#121—3621no—
245#121—3621no—
246#121C2×D18111yescharacteristic
247C4×Q8—1811no—
248C4×D4—3621no—
249C4×D4—1811no—
250C4×D4—1811no—
251C2×C4⋊C4—3621no—
252#107—1811no—
253C4⋊C8—1811no—
254#103—1811no—
255#103—1811no—
256#103—1811no—
257#103C2×D18111yescharacteristic
258#102—1811no—
259#102C2×D18111yescharacteristic
260C22⋊C8—1811no—
261C22⋊C8C2×D18111yescharacteristic
262#97—1811no—
263#41—22no—
264#41—41no—
265#41—3622no—
266#41—3622no—
267#41—7241no—
268#41—7241no—
269C2×Q8—3622no—
270C2×Q8—3621no—
271C2×Q8—1811no—
272C2×Q8—1811no—
273C2×Q8—1811no—
274C2×Q8—7241no—
275C2×Q8C22×D18111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3621no—
279C2×D4—3621no—
280C2×D4—1811no—
281C2×D4C22×D18111yescharacteristic
282C2×D4C22×D18111yescharacteristic
283C22×C4—21no—
284C22×C4—3621no—
285C22×C4—3621no—
286C22×C4—3621no—
287C22×C4—3621no—
288C22×C4—1811no—
289C22×C4C22×D18111yescharacteristic
290Q16—3622no—
291Q16—3622no—
292Q16—3622no—
293Q16—7241no—
294SD16—41no—
295SD16—3622no—
296SD16—3622no—
297SD16—1814no—
298D8—22no—
299M16—3622no—
300C2×C8—3622no—
301C2×C8—1811no—
302C2×C8—1811no—
303C2×C8C22×D18111yescharacteristic
304C2×C8C22×D18111yescharacteristic
305C4⋊C4—3621no—
306C4⋊C4—3621no—
307C4⋊C4—3621no—
308C4⋊C4—3621no—
309C4⋊C4—3621no—
310C4⋊C4—3621no—
311C4⋊C4—1811no—
312C4⋊C4—1811no—
313C4⋊C4C22×D18111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—3621no—
316C22⋊C4—3621no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—3621no—
321C22⋊C4—1811no—
322C22⋊C4—1811no—
323C42—3621no—
324C42—1811no—
325C23—21no—
326C23—21no—
327C23—1811no—
328C23D4×D18111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—3621no—
332Q8—3621no—
333Q8—3621no—
334Q8—3621no—
335Q8—3621no—
336Q8—1812no—
337Q8—7242no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—7241no—
344D4—7242no—
345D4—7242no—
346D4D4×D18112yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—3621no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—3621no—
358C2×C4—1811no—
359C2×C4—1811no—
360C2×C4—1811no—
361C2×C4—7242no—
362C2×C4—7242no—
363C2×C4—7242no—
364C2×C4—7242no—
365C2×C4D4×D18111yescharacteristic
366C2×C4C23×D18111yescharacteristic
367C8—21no—
368C8—21no—
369C8—3621no—
370C8—1812no—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3621no—
381C22—3621no—
382C22—3621no—
383C22C2×D4×D18111yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—3621no—
388C4—3622no—
389C4—3621no—
390C4—3621no—
391C4—7241no—
392C4—7241no—
393C4—7241no—
394C4C2×D4×D18111yescharacteristic
395C4#1915386111yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—7241no—
400C2#2604191111yescharacteristic
401C2#2604207511yescharacteristic
402C2#2604209311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#449×C181C2
socleC22×C181C22×D4
hypercenter#449D181
hypocenterC181#3269
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19153846C2semidirect
2. C2×#19153835C2semidirect
3. #26041935C2semidirect
4. #26041928C2semidirect
5. #26041670C2semidirect
6. #26041523C2semidirect
7. #26041519C2semidirect
8. #26041492C2semidirect
9. #26041488C2semidirect
10. #26041442C2semidirect
11. #26041438C2semidirect
12. #26041437C2semidirect
13. #26041376C2semidirect
14. #26041353C2semidirect
15. #449×C181C2semidirect
85. #15668437D4semidirect
201. C181#3269semidirect
215. #449D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538462
C2×#191538352
#260419352
#260419282
#260416702
#260415232
#260415192
#260414922
#260414882
#260414422
#260414382
#260414372
#260413762
#260413532
#449×C1812
#3269181