Finite Group Databasebeta

← #37544575

Subgroups — #37544575

Order 23168.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041857C211yesmaximal, characteristic, complemented
2#26041843C211yesmaximal, characteristic, complemented
3#26041621C211yesmaximal, characteristic, complemented
4#26041604C211yesmaximal, characteristic, complemented
5#26041598C211yesmaximal, characteristic, complemented
6#26041593C211yesmaximal, characteristic, complemented
7#26041532C211yesmaximal, characteristic, complemented
8#26041525C211yesmaximal, characteristic, complemented
9#26041515C211yesmaximal, characteristic, complemented
10#26041508C211yesmaximal, characteristic, complemented
11#26041432C211yesmaximal, characteristic, complemented
12#26041426C211yesmaximal, characteristic, complemented
13#26041426C211yesmaximal, characteristic, complemented
14#26041415C211yesmaximal, characteristic, complemented
15#408×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic, complemented
17C22×D724C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×D1448C2211yescharacteristic
21C2×D1448C2211yescharacteristic
22#19153812—21no—
23#19153805—21no—
24#19153795C2211yescharacteristic
25#19153795C2211yescharacteristic, complemented
26#19153792—21no—
27#19153790C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153790C2211yescharacteristic
30#19153790C2211yescharacteristic
31#19153790C2211yescharacteristic
32#19153790C2211yescharacteristic, complemented
33#19153788—21no—
34#19153788—21no—
35#19153787—21no—
36#19153784—21no—
37#19153784—21no—
38#19153775C2211yescharacteristic
39C2×C4⋊Dic181C2211yescharacteristic, complemented
40#19153750C2211yescharacteristic, complemented
41#19153750C2211yescharacteristic, complemented
42#19153748C2211yescharacteristic, complemented
43#19153748C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153741C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153738C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic, complemented
52C2×#41×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic, complemented
54C22×C8×C181C2211yescharacteristic, complemented
55C22⋊C8×C181C2211yescharacteristic
56C22⋊C8×C181C2211yescharacteristic
57C22⋊C8×C181C2211yescharacteristic, complemented
58C22⋊C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668449—41no—
61#15668449—41no—
62C2×Dic362C2311yescharacteristic
63C2×D724—21no—
64C2×D724—41no—
65C2×D724—41no—
66C2×D724C2311yescharacteristic
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C22×Dic181—21no—
77C2×C4×D181—21no—
78#15668438—41no—
79#15668438—41no—
80D1448—41no—
81D1448—41no—
82C4⋊Dic181—21no—
83C4⋊Dic181C2311yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86C4⋊Dic181C2311yescharacteristic
87#15668428—21no—
88#15668428—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98C4×Dic181—21no—
99#41×C181—41no—
100#41×C181—41no—
101C2×Q8×C181D411yescharacteristic
102C2×D4×C181D411yescharacteristic
103C2×D4×C181D411yescharacteristic
104C2×D4×C181D411yescharacteristic
105C22×C4×C181C2311yescharacteristic
106C22×C4×C181C2311yescharacteristic
107C22×C4×C181C2311yescharacteristic
108M16×C181—41no—
109C2×C8×C181—41no—
110C2×C8×C181C2311yescharacteristic
111C2×C8×C181C2311yescharacteristic
112C2×C8×C181C2311yescharacteristic
113C2×C8×C181C2311yescharacteristic
114C22×D181—21no—
115C22×D181—21no—
116C22×D181—21no—
117C22×D181—42no—
118C22×D181—81no—
119C22×D181—81no—
120Dic362—21no—
121Dic362—21no—
122D724—21no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724—21no—
127D724—21no—
128D724—42no—
129C22⋊D181—42no—
130C22⋊D181—81no—
131C22⋊D181—81no—
132C22⋊D181—81no—
133C22⋊D181—81no—
134C22⋊D181—81no—
135C22⋊D181—81no—
136C2×Dic181—21no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—21no—
140C2×Dic181—21no—
141C2×Dic181—81no—
142C4×D181—42no—
143C23×C181C2×D411yescharacteristic
144C23×C181C2×D411yescharacteristic
145C23×C181C2×D411yescharacteristic
146Q8×C181—41no—
147D4×C181—41no—
148D4×C181—41no—
149D4×C181—41no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2411yescharacteristic
160C8×C181—21no—
161C8×C181—21no—
162C8×C181—21no—
163C8×C181—21no—
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181C22×D411yescharacteristic
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—41no—
178C2×D181—41no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—84no—
184C4×C181—21no—
185C4×C181—21no—
186C4×C181—21no—
187C4×C181H(5,𝔽2)11yescharacteristic
188C4×C181H(5,𝔽2)11yescharacteristic
189Dic181—41no—
190Dic181—41no—
191Dic181—41no—
192Dic181—41no—
193Dic181—41no—
194C2×C181—41no—
195C2×C181—41no—
196C2×C181—41no—
197C2×C181#53411yescharacteristic
198C2×C181#57611yescharacteristic
199C2×C181#57711yescharacteristic
200D181—81no—
201D181—81no—
202D181—81no—
203C181#319811yescharacteristic, complemented, Sylow
204#3198—1811nomaximal, Sylow
205#537—1811no—
206#535—1811no—
207#485—1811no—
208#482—1811no—
209#481—1811no—
210#480—1811no—
211#469—1811no—
212#468—1811no—
213#466—1811no—
214#465—1811no—
215#450—1811no—
216#449—1811no—
217#449—1811no—
218#447—1811no—
219#408D18111yescharacteristic, complemented
220C2×#41—1811no—
221C2×#41C2×D18111yescharacteristic
222C22×D4—1811no—
223C2×SD16—1811no—
224C2×SD16—1811no—
225C2×D8—1811no—
226C2×D8—1811no—
227C2×M16C2×D18111yescharacteristic, complemented
228C22×C8C2×D18111yescharacteristic, complemented
229#127—3621no—
230#124—3621no—
231C22⋊Q8—3621no—
232C22⋊Q8—1811no—
233C22⋊Q8—1811no—
234#121—3621no—
235#121—3621no—
236#121—3621no—
237#121—1811no—
238#121—1811no—
239#121—1811no—
240#121—1811no—
241#121—1811no—
242#121—1811no—
243#120—3621no—
244#120—3621no—
245#117—1811no—
246C2×C4⋊C4—1811no—
247#107—1811no—
248#107—1811no—
249#106—1811no—
250#106—1811no—
251#103—1811no—
252#103—1811no—
253#102—1811no—
254#102—1811no—
255#102—1811no—
256#102—1811no—
257#102—1811no—
258#102—1811no—
259C22⋊C8C2×D18111yescharacteristic
260C22⋊C8C2×D18111yescharacteristic
261C22⋊C8C2×D18111yescharacteristic, complemented
262C22⋊C8C2×D18111yescharacteristic, complemented
263C24—3621no—
264#41—41no—
265#41—41no—
266#41—7241no—
267#41—7241no—
268C2×Q8—1811no—
269C2×Q8D72411yescharacteristic
270C2×D4—3621no—
271C2×D4—3621no—
272C2×D4—3621no—
273C2×D4—3621no—
274C2×D4—3621no—
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—3621no—
278C2×D4—1811no—
279C2×D4—1811no—
280C2×D4—1811no—
281C2×D4—7241no—
282C2×D4—7241no—
283C2×D4D72411yescharacteristic
284C2×D4D72411yescharacteristic
285C2×D4D72411yescharacteristic
286C22×C4—3621no—
287C22×C4—3621no—
288C22×C4C22×D18111yescharacteristic
289C22×C4C22×D18111yescharacteristic
290C22×C4C22×D18111yescharacteristic
291SD16—7241no—
292SD16—7241no—
293D8—7241no—
294D8—7241no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D18111yescharacteristic
298C2×C8C22×D18111yescharacteristic
299C2×C8C22×D18111yescharacteristic
300C2×C8C22×D18111yescharacteristic
301C4⋊C4—3621no—
302C4⋊C4—3621no—
303C4⋊C4—3621no—
304C4⋊C4—1811no—
305C4⋊C4—1811no—
306C4⋊C4—1811no—
307C4⋊C4—1811no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C22⋊C4—3621no—
311C22⋊C4—3621no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—3621no—
317C42—3621no—
318C23—3621no—
319C23—3621no—
320C23—3621no—
321C23—14481no—
322C23—14481no—
323C23—7242no—
324C23C2×D72411yescharacteristic
325C23C2×D72411yescharacteristic
326C23C2×D72411yescharacteristic
327Q8—41no—
328Q8—3621no—
329Q8—3621no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—3621no—
334D4—3621no—
335D4—3621no—
336D4—3621no—
337D4—3621no—
338D4—3621no—
339D4—14481no—
340D4—14481no—
341D4—14481no—
342D4—14481no—
343D4—14481no—
344D4—14481no—
345D4—7242no—
346D4—7242no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3621no—
354C2×C4—3621no—
355C2×C4—3621no—
356C2×C4—3621no—
357C2×C4—3621no—
358C2×C4—14481no—
359C2×C4—7242no—
360C2×C4C2×D72411yescharacteristic
361C2×C4C2×D72411yescharacteristic
362C2×C4C2×D72411yescharacteristic
363C2×C4C23×D18111yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—14484no—
378C22—7241no—
379C22—7241no—
380C22—7241no—
381C22—7241no—
382C22—7241no—
383C22—7241no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22C22×D72411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—7241no—
392C4—7241no—
393C4—7241no—
394C4—7241no—
395C4—7241no—
396C4#1915386111yescharacteristic
397C4#1915386111yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—14481no—
402C2—14481no—
403C2—14481no—
404C2#2604184111yescharacteristic
405C2#2604208411yescharacteristic
406C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#408×C181C2
socleC22×C181C22×D4
hypercenter#408D181
hypocenterC181#3198
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #26041857C2semidirect
2. #26041843C2semidirect
3. #26041621C2semidirect
4. #26041604C2semidirect
5. #26041598C2semidirect
6. #26041593C2semidirect
7. #26041532C2semidirect
8. #26041525C2semidirect
9. #26041515C2semidirect
10. #26041508C2semidirect
11. #26041432C2semidirect
12. #26041426C2semidirect
13. #26041426C2semidirect
14. #26041415C2semidirect
15. #408×C181C2semidirect
16. C2×#15668449C22semidirect
25. #19153795C22semidirect
32. #19153790C22semidirect
39. C2×C4⋊Dic181C22semidirect
40. #19153750C22semidirect
41. #19153750C22semidirect
42. #19153748C22semidirect
43. #19153748C22semidirect
44. #19153741C22semidirect
45. #19153741C22semidirect
50. #19153738C22semidirect
51. #19153738C22semidirect
53. C2×M16×C181C22semidirect
54. C22×C8×C181C22semidirect
57. C22⋊C8×C181C22semidirect
58. C22⋊C8×C181C22semidirect
203. C181#3198semidirect
219. #408D181semidirect
227. C2×M16C2×D181semidirect
228. C22×C8C2×D181semidirect
261. C22⋊C8C2×D181semidirect
262. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418572
#260418432
#260416212
#260416042
#260415982
#260415932
#260415322
#260415252
#260415152
#260415082
#260414322
#260414262
#260414152
#408×C1812
#3198181