Finite Group Databasebeta

← #37545455

Subgroups — #37545455

Order 23168.

482 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×#19153837C211yesmaximal, characteristic, complemented
2C22×#15668439C211yesmaximal, characteristic, complemented
3#26041940C211yesmaximal, characteristic, complemented
4#26041935C211yesmaximal, characteristic, complemented
5#26041660C211yesmaximal, characteristic, complemented
6#26041519C211yesmaximal, characteristic, complemented
7#26041513C211yesmaximal, characteristic, complemented
8#26041504C211yesmaximal, characteristic, complemented
9#26041499C211yesmaximal, characteristic, complemented
10#26041444C211yesmaximal, characteristic, complemented
11#26041438C211yesmaximal, characteristic, complemented
12#26041420C211yesmaximal, characteristic, complemented
13#26041371C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#448×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C22×Dic362C2211yescharacteristic
19#19153837—24no—
20C2×#15668439—24no—
21C2×#15668439C2211yescharacteristic
22C2×#15668439C2211yescharacteristic
23C2×#15668439C2211yescharacteristic
24C2×#15668439C2211yescharacteristic
25C2×#15668438C2211yescharacteristic
26C2×#15668438C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C22×#13089678C2211yescharacteristic
29C2×C8×D181C2211yescharacteristic
30#19153805—21no—
31#19153805C2211yescharacteristic
32#19153803C2211yescharacteristic
33#19153802—21no—
34#19153796C2211yescharacteristic
35#19153793—21no—
36#19153792—21no—
37#19153792C2211yescharacteristic
38#19153790—21no—
39#19153790C2211yescharacteristic
40#19153788—21no—
41Q8×Dic181C2211yescharacteristic
42#19153781C2211yescharacteristic
43D4×Dic181—21no—
44C4×C22⋊D181—21no—
45C4×C22⋊D181C2211yescharacteristic
46#19153749C2211yescharacteristic
47#19153745C2211yescharacteristic
48#19153741C2211yescharacteristic
49#19153741C2211yescharacteristic
50#19153737C2211yescharacteristic
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53C8×Dic181C2211yescharacteristic
54C22×Q8×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic
56C2×SD16×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#103×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic
61#15668450—22no—
62#15668450—41no—
63#15668448—22no—
64#15668448—41no—
65C2×Dic362—22no—
66C2×Dic362—21no—
67C2×Dic362—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×D724—21no—
71C2×D724C2311yescharacteristic
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181D411yescharacteristic
75C22×Dic181—21no—
76C22×Dic181—21no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79#15668439—22no—
80#15668439—22no—
81#15668439—22no—
82#15668439D414yescomplemented
83#15668438—22no—
84#15668438—41no—
85#15668434—22no—
86C2×#13089678—22no—
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C8×D181—22no—
90C4⋊Dic181—21no—
91C4⋊Dic181—21no—
92C4⋊Dic181C2311yescharacteristic
93#15668428—21no—
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181D411yescharacteristic
99#15668426—21no—
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426D411yescharacteristic
105C4×Dic181—21no—
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108C2×Q8×C181—21no—
109C2×Q8×C181—22no—
110C2×Q8×C181—41no—
111C2×Q8×C181C2311yescharacteristic
112C2×Q8×C181C2311yescharacteristic
113C2×D4×C181—21no—
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—22no—
118SD16×C181—41no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181C2×D411yescharacteristic
125Dic362—21no—
126Dic362—21no—
127Dic362—21no—
128Dic362—42no—
129Dic362C2×D412yes—
130D724—21no—
131D724—21no—
132D724—42no—
133C22⋊D181—22no—
134C22⋊D181—42no—
135C22⋊D181—42no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—42no—
140C2×Dic181—42no—
141C2×Dic181C2×D411yescharacteristic
142C2×Dic181C2×D411yescharacteristic
143C2×Dic181C2×D411yescharacteristic
144C4×D181—21no—
145C4×D181—21no—
146C4×D181—42no—
147#13089678—21no—
148#13089678C2×D412yes—
149C23×C181—21no—
150C23×C181C2×D411yescharacteristic
151Q8×C181—21no—
152Q8×C181—21no—
153Q8×C181—21no—
154Q8×C181—42no—
155Q8×C181C2×D412yes—
156D4×C181—21no—
157D4×C181—21no—
158D4×C181—42no—
159C2×C4×C181—21no—
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—21no—
171C22×C181—41no—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181#13512yescomplemented
175C22×C181C22×D411yescharacteristic
176C2×D181—21no—
177C2×D181—21no—
178C2×D181—21no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C4×C181—21no—
183C4×C181—22no—
184C4×C181—41no—
185C4×C181—41no—
186C4×C181C22×D411yescharacteristic
187C4×C181H(5,𝔽2)11yescharacteristic
188Dic181—21no—
189Dic181—22no—
190Dic181—21no—
191Dic181—21no—
192Dic181—41no—
193Dic181—41no—
194C2×C181—22no—
195C2×C181—81no—
196C2×C181D4211yescharacteristic
197C2×C181C2×#13511yescharacteristic
198C2×C181#57811yescharacteristic
199D181—41no—
200D181—81no—
201C181#328511yescharacteristic, complemented, Sylow
202#3285—1811nomaximal, Sylow
203C2×#135—1811no—
204C22×Q16—1811no—
205#548—1811no—
206#548—1811no—
207#495—1811no—
208#467—1811no—
209#466—1811no—
210#464—1811no—
211#464—1811no—
212#452—1811no—
213#452—1811no—
214#448—1811no—
215#448D18111yescharacteristic, complemented
216C4×Q16—1811no—
217C8×D4—1811no—
218C2×#41—1811no—
219C2×#41—1811no—
220C22×Q8—1811no—
221C22×Q8C2×D18111yescharacteristic
222#135—3624no—
223C2×Q16—3624no—
224C2×Q16—1811no—
225C2×Q16—1811no—
226C2×Q16—1811no—
227C2×Q16—1811no—
228C2×SD16—1811no—
229C2×SD16—1811no—
230C2×SD16C2×D18111yescharacteristic
231C2×SD16C2×D18111yescharacteristic
232C2×D8—1811no—
233C22×C8—1811no—
234C22×C8—1811no—
235#124—3621no—
236#124—3621no—
237#124—1811no—
238#124—1811no—
239C22⋊Q8—3621no—
240C22⋊Q8—3621no—
241C22⋊Q8—1811no—
242C22⋊Q8—1811no—
243#121—3621no—
244#121—3621no—
245#121—1811no—
246#121C2×D18111yescharacteristic
247C4×Q8—1811no—
248C4×Q8—1811no—
249C4×D4—3621no—
250C4×D4—3621no—
251C4×D4—1811no—
252#107—1811no—
253C4⋊C8—1811no—
254#103—1811no—
255#103—1811no—
256#103C2×D18111yescharacteristic
257#103C2×D18111yescharacteristic
258#102—1811no—
259#102—1811no—
260C22⋊C8—1811no—
261C22⋊C8C2×D18111yescharacteristic
262C4×C8—1811no—
263#41—3622no—
264#41—3622no—
265#41—7241no—
266#41—7241no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—3622no—
271C2×Q8—3621no—
272C2×Q8—1811no—
273C2×Q8—1811no—
274C2×Q8—7241no—
275C2×Q8C22×D18111yescharacteristic
276C2×Q8C22×D18111yescharacteristic
277C2×D4—21no—
278C2×D4—3621no—
279C2×D4—3621no—
280C2×D4—3621no—
281C2×D4—1811no—
282C2×D4—1811no—
283C2×D4C22×D18111yescharacteristic
284C22×C4—21no—
285C22×C4—3621no—
286C22×C4—3621no—
287C22×C4—3621no—
288C22×C4—1811no—
289C22×C4C22×D18111yescharacteristic
290Q16—3622no—
291Q16—3622no—
292Q16—3622no—
293Q16—1814no—
294SD16—22no—
295SD16—41no—
296SD16—3622no—
297SD16—7241no—
298D8—3622no—
299C2×C8—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—1811no—
310C4⋊C4—1811no—
311C4⋊C4C22×D18111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—3621no—
317C22⋊C4—3621no—
318C22⋊C4—3621no—
319C22⋊C4—3621no—
320C22⋊C4—1811no—
321C22⋊C4—1811no—
322C42—3621no—
323C42—3621no—
324C42—1811no—
325C23—21no—
326C23—3621no—
327C23—1811no—
328C23D4×D18111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3621no—
334Q8—3621no—
335Q8—3621no—
336Q8—1812no—
337Q8—7242no—
338Q8D4×D18112yes—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—3622no—
343D4—3621no—
344D4—3621no—
345D4—7242no—
346D4—7242no—
347D4—7242no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—3621no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—3621no—
358C2×C4—3621no—
359C2×C4—1811no—
360C2×C4—1811no—
361C2×C4—1811no—
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—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—3621no—
377C22—3621no—
378C22—3621no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22#1915383712yescomplemented
383C22C2×D4×D18111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—3621no—
389C4—3622no—
390C4—3621no—
391C4—3621no—
392C4—7241no—
393C4—7241no—
394C4C2×D4×D18111yescharacteristic
395C4#1915386111yescharacteristic
396C2—22no—
397C2—81no—
398C2—14481no—
399C2—7241no—
400C2#2604191111yescharacteristic
401C2C2×#1915383711yescharacteristic
402C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#448×C181C2
socleC22×C181C22×D4
hypercenter#448D181
hypocenterC181#3285
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19153837C2semidirect
2. C22×#15668439C2semidirect
3. #26041940C2semidirect
4. #26041935C2semidirect
5. #26041660C2semidirect
6. #26041519C2semidirect
7. #26041513C2semidirect
8. #26041504C2semidirect
9. #26041499C2semidirect
10. #26041444C2semidirect
11. #26041438C2semidirect
12. #26041420C2semidirect
13. #26041371C2semidirect
14. #26041342C2semidirect
15. #448×C181C2semidirect
82. #15668439D4semidirect
174. C22×C181#135semidirect
201. C181#3285semidirect
215. #448D181semidirect
382. C22#19153837semidirect

Maximal subgroups

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

subgroupindex
C2×#191538372
C22×#156684392
#260419402
#260419352
#260416602
#260415192
#260415132
#260415042
#260414992
#260414442
#260414382
#260414202
#260413712
#260413422
#448×C1812
#3285181