Finite Group Databasebeta

← #28404220

Subgroups — #28404220

Order 14464.

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×#16813669C211yesmaximal, characteristic, complemented
2C22×Q8×D113C211yesmaximal, characteristic, complemented
3Q8×D452C211yesmaximal, characteristic, complemented
4D4×Dic226C213yesmaximal, complemented
5Q8×C22⋊D113C211yesmaximal, characteristic, complemented
6#20299261C213yesmaximal, complemented
7#20299256C213yesmaximal, complemented
8#20299255C211yesmaximal, characteristic, complemented
9#20299254C213yesmaximal, complemented
10#20299171C213yesmaximal, complemented
11#20299168C213yesmaximal, complemented
12#20299108C213yesmaximal, complemented
13#20299107C213yesmaximal, complemented
14#20299074C211yesmaximal, characteristic, complemented
15#548×C113C211yesmaximal, characteristic, complemented
16#16813669—28no—
17C2×#14268647C2213yes—
18C2×#14268646C2213yes—
19C2×#14268646C2213yescomplemented
20C22×Dic226C2213yes—
21C22×Dic226C2213yescomplemented
22C2×Q8×D113—28no—
23C2×Q8×D113C2211yescharacteristic
24C2×Q8×D113C2213yes—
25C2×Q8×D113C2213yes—
26C2×Q8×D113C2211yescharacteristic
27C2×Q8×D113C2211yescharacteristic, complemented
28C22×C4×D113C2213yes—
29#16813622C2213yescomplemented
30#16813621C2216yescomplemented
31#16813620C2213yescomplemented
32#16813609C2213yescomplemented
33#16813608C2216yescomplemented
34#16813607C2213yes—
35#16813604C2213yes—
36#16813604C2216yes—
37#16813602C2213yes—
38#16813602C2213yescomplemented
39#16813601C2213yes—
40#16813601C2213yescomplemented
41#16813600C2213yes—
42#16813599C2213yes—
43#16813599C2216yescomplemented
44#16813599C2216yescomplemented
45#16813598C2213yescomplemented
46#16813595C2213yes—
47#16813594C2213yes—
48#16813593C2213yes—
49#16813592C2211yescharacteristic
50#16813590C2213yes—
51Q8×Dic113C2211yescharacteristic, complemented
52C4×Dic226C2213yescomplemented
53#16813587C2213yescomplemented
54C4×D452C2213yes—
55D4×Dic113C2216yes—
56C4×C22⋊D113C2213yes—
57C4×C22⋊D113C2213yes—
58#16813582C2216yes—
59#16813580C2213yes—
60#16813579C2213yes—
61C2×#41×C113C2211yescharacteristic, complemented
62C22×Q8×C113C2211yescharacteristic
63#124×C113C2213yescomplemented
64C22⋊Q8×C113C2213yescomplemented
65#121×C113C2213yes—
66C4×Q8×C113C2211yescharacteristic, complemented
67C4×D4×C113C2213yes—
68C23×D113C2311yescharacteristic
69#14268647—212no—
70#14268646—212no—
71#14268646—212no—
72C2×Dic226—212no—
73C2×Dic226—212no—
74C2×Dic226C2313yes—
75C2×Dic226C2316yes—
76C2×Dic226C2313yes—
77C2×Dic226C2313yes—
78C2×Dic226C2313yes—
79Q8×D113—24no—
80Q8×D113—24no—
81Q8×D113—212no—
82Q8×D113—212no—
83Q8×D113D418yescomplemented
84C2×D452C2311yescharacteristic
85C2×D452C2311yescharacteristic
86C2×C22⋊D113C2313yes—
87C2×C22⋊D113C2311yescharacteristic
88C2×C22⋊D113C2313yes—
89C2×C22⋊D113C2313yes—
90C22×Dic113C2316yes—
91C22×Dic113C2313yes—
92C2×C4×D113—212no—
93C2×C4×D113C2313yes—
94C2×C4×D113C2313yes—
95C2×C4×D113C2313yes—
96C2×C4×D113C2313yes—
97C2×C4×D113C2313yes—
98C4⋊Dic113C2313yes—
99C4⋊Dic113C2316yes—
100C4⋊Dic113C2313yes—
101#14268626C2313yes—
102#14268626C2313yes—
103#14268626C2316yes—
104#14268626C2313yes—
105#14268626C2313yes—
106C22⋊Dic113C2313yes—
107C22⋊Dic113C2313yes—
108C22⋊Dic113C2316yes—
109#14268624C2311yescharacteristic
110#14268624C2313yes—
111#14268624C2313yes—
112#14268624C2311yescharacteristic
113#14268624C2313yes—
114#14268624C2313yes—
115C4×Dic113C2313yes—
116C4×Dic113C2316yes—
117#41×C113—24no—
118C2×Q8×C113—24no—
119C2×Q8×C113C2313yes—
120C2×Q8×C113C2311yescharacteristic
121C2×D4×C113C2313yes—
122C2×D4×C113C2313yes—
123C22×C4×C113C2313yes—
124C22×C4×C113C2313yes—
125C4⋊C4×C113C2313yes—
126C4⋊C4×C113C2313yes—
127C22⋊C4×C113C2316yes—
128C22⋊C4×C113C2311yescharacteristic
129C22⋊C4×C113C2313yes—
130C42×C113C2313yes—
131C23×C113C2413yes—
132C23×C113C2411yescharacteristic
133C22×D113—22no—
134C22×D113—22no—
135C22×D113—22no—
136C22×D113C2411yescharacteristic
137C22×D113C2411yescharacteristic
138C22×D113C2411yescharacteristic
139Q8×C113—26no—
140Q8×C113C2×D414yes—
141D4×C113—26no—
142D4×C113—26no—
143C2×C4×C113—26no—
144C2×C4×C113—26no—
145C2×C4×C113C2411yescharacteristic
146C2×C4×C113C2413yes—
147C2×C4×C113C2413yes—
148C2×C4×C113C2411yescharacteristic
149C2×C4×C113C2413yes—
150C22⋊D113—22no—
151C22⋊D113—26no—
152C22⋊D113—26no—
153C22⋊D113—26no—
154C2×Dic113—26no—
155C2×Dic113—212no—
156C2×Dic113C2411yescharacteristic
157C2×Dic113C2413yes—
158C2×Dic113C2413yes—
159C2×Dic113C2413yes—
160C2×Dic113C2413yes—
161D452—22no—
162D452—22no—
163C4×D113—26no—
164C4×D113—26no—
165C4×D113—26no—
166C4×D113—26no—
167C4×D113C2×D4112yes—
168Dic226—26no—
169Dic226—26no—
170Dic226—212no—
171Dic226—26no—
172Dic226C2×D4112yes—
173C22×C113—21no—
174C22×C113—21no—
175C22×C113—23no—
176C22×C113—23no—
177C22×C113—23no—
178C22×C1132−1+412yescomplemented
179C22×C113C2511yescharacteristic
180C2×D113—28no—
181C2×D113—21no—
182C2×D113—21no—
183C2×D113—21no—
184C2×D113—21no—
185C2×D113—21no—
186C2×D113—21no—
187C2×D113—21no—
188C2×D113C22×D412yes—
189C2×D1132−1+412yescomplemented
190C4×C113—21no—
191C4×C113—23no—
192C4×C113—21no—
193C4×C113—23no—
194C4×C113C22×D416yes—
195Dic113—23no—
196Dic113—23no—
197Dic113—23no—
198Dic113—21no—
199Dic113C22×D416yes—
200C2×C113—22no—
201C2×C113—43no—
202C2×C113C23×D411yescharacteristic
203C2×C113C2×2−1+411yescharacteristic
204C2×C113C2×2−1+411yescharacteristic
205D113—22no—
206D113—22no—
207D113—41no—
208#3486—1131nomaximal, Sylow
209C113#348611yescharacteristic, complemented, Sylow
210C2×2−1+4—1131no—
211C23×Q8—1131no—
212D4×Q8—1131no—
213D4×Q8—1133no—
214D4×Q8—1131no—
215D4×Q8—1133no—
216#548—1131no—
217#548—1133no—
218#548—1133no—
219#548D11311yescharacteristic, complemented
220#536—1133no—
221#536—1133no—
222#525—1133no—
223#525—1133no—
224#519—1131no—
2252−1+4—2268no—
226C2×#41—1133no—
227C2×#41—1133no—
228C2×#41—1133no—
229C2×#41C2×D11311yescharacteristic, complemented
230C22×Q8—1131no—
231C22×Q8—1133no—
232C22×Q8—1133no—
233C22×Q8—1131no—
234C22×Q8—1131no—
235C22×Q8—1133no—
236C22×Q8—1133no—
237C22×Q8—2268no—
238C22×Q8C2×D11311yescharacteristic
239C23×C4—1133no—
240C4⋊Q8—1133no—
241C4⋊Q8—1136no—
242C4⋊Q8—1133no—
243#124—1133no—
244#124—1136no—
245#124C2×D11313yescomplemented
246#123—1133no—
247#123—1136no—
248#123—1133no—
249C22⋊Q8—1133no—
250C22⋊Q8—1133no—
251C22⋊Q8—1133no—
252C22⋊Q8—1133no—
253C22⋊Q8—1133no—
254C22⋊Q8—1133no—
255C22⋊Q8—1133no—
256C22⋊Q8—1136no—
257C22⋊Q8—1136no—
258C22⋊Q8C2×D11313yescomplemented
259#121—1133no—
260#121—1133no—
261#121—1133no—
262#121C2×D11313yes—
263#120—1133no—
264#120—1131no—
265C4×Q8—1131no—
266C4×Q8—1133no—
267C4×Q8—1133no—
268C4×Q8C2×D11311yescharacteristic, complemented
269C4×D4—1136no—
270C4×D4—1133no—
271C4×D4—1133no—
272C4×D4—1136no—
273C4×D4—1133no—
274C4×D4C2×D11313yes—
275#117—1133no—
276#117—1133no—
277C24—1131no—
278#41—24no—
279#41—22612no—
280#41—22612no—
281#41—22612no—
282C2×Q8—24no—
283C2×Q8—1138no—
284C2×Q8—1133no—
285C2×Q8—1136no—
286C2×Q8—1133no—
287C2×Q8—1133no—
288C2×Q8—1133no—
289C2×Q8—22612no—
290C2×Q8—22612no—
291C2×Q8—22612no—
292C2×Q8—22612no—
293C2×Q8—2264no—
294C2×Q8—2264no—
295C2×Q8C22×D11313yes—
296C2×Q8C22×D11311yescharacteristic
297C2×D4—1133no—
298C2×D4—1131no—
299C2×D4—1133no—
300C2×D4—1133no—
301C2×D4—1131no—
302C2×D4—1131no—
303C2×D4C22×D11313yes—
304C2×D4C22×D11313yes—
305C22×C4—1133no—
306C22×C4—1133no—
307C22×C4—1136no—
308C22×C4—1133no—
309C22×C4—1133no—
310C22×C4—1133no—
311C22×C4—1133no—
312C22×C4—22612no—
313C22×C4C22×D11313yes—
314C22×C4C22×D11313yes—
315C4⋊C4—1133no—
316C4⋊C4—1133no—
317C4⋊C4—1133no—
318C4⋊C4—1133no—
319C4⋊C4—1133no—
320C4⋊C4—1136no—
321C4⋊C4—1136no—
322C4⋊C4—1133no—
323C4⋊C4C22×D11313yes—
324C4⋊C4C22×D11313yes—
325C22⋊C4—1133no—
326C22⋊C4—1136no—
327C22⋊C4—1133no—
328C22⋊C4—1133no—
329C22⋊C4—1131no—
330C22⋊C4—1131no—
331C22⋊C4—1133no—
332C22⋊C4—1133no—
333C22⋊C4—1133no—
334C22⋊C4C22×D11316yes—
335C22⋊C4C22×D11311yescharacteristic
336C22⋊C4C22×D11313yes—
337C42—1136no—
338C42—1133no—
339C42C22×D11313yes—
340C23—1131no—
341C23—1131no—
342C23—1131no—
343C23—2262no—
344C23—2262no—
345C23—2262no—
346C23C23×D11313yes—
347C23C23×D11311yescharacteristic
348Q8—26no—
349Q8—11312no—
350Q8—22612no—
351Q8—2266no—
352Q8—2266no—
353Q8—2266no—
354Q8D4×D11314yes—
355D4—26no—
356D4—26no—
357D4—2262no—
358D4—2262no—
359D4—2266no—
360D4—2266no—
361D4—2266no—
362D4—2262no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—1133no—
366C2×C4—11312no—
367C2×C4—1133no—
368C2×C4—1131no—
369C2×C4—1133no—
370C2×C4—1133no—
371C2×C4—2266no—
372C2×C4—2266no—
373C2×C4—2266no—
374C2×C4—22612no—
375C2×C4—2266no—
376C2×C4—2266no—
377C2×C4C23×D11313yes—
378C2×C4C23×D11311yescharacteristic
379C2×C4C23×D11313yes—
380C2×C4C23×D11311yescharacteristic
381C2×C4C23×D11313yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—1132no—
388C22—1132no—
389C22—2268no—
390C22—2261no—
391C22—2261no—
392C22—2261no—
393C22—2261no—
394C22—2261no—
395C22—2261no—
396C22—2261no—
397C22#1681366912yescomplemented
398C22C24×D11311yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—1136no—
404C4—2263no—
405C4—2263no—
406C4—2263no—
407C4—2261no—
408C4C2×D4×D11316yes—
409C2—22no—
410C2—43no—
411C2—4521no—
412C2—2262no—
413C2—2262no—
414C2C22×D4×D11311yescharacteristic
415C22−1+4×D11311yescharacteristic
416C2C2×#1681366911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D113
commutator, socle, cosocleC22×C113C25
Fitting#548×C113C2
hypercenter#548D113
hypocenterC113#3486
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16813669C2semidirect
2. C22×Q8×D113C2semidirect
3. Q8×D452C2semidirect
4. D4×Dic226C2semidirect
5. Q8×C22⋊D113C2semidirect
6. #20299261C2semidirect
7. #20299256C2semidirect
8. #20299255C2semidirect
9. #20299254C2semidirect
10. #20299171C2semidirect
11. #20299168C2semidirect
12. #20299108C2semidirect
13. #20299107C2semidirect
14. #20299074C2semidirect
15. #548×C113C2semidirect
19. C2×#14268646C22semidirect
21. C22×Dic226C22semidirect
27. C2×Q8×D113C22semidirect
29. #16813622C22semidirect
30. #16813621C22semidirect
31. #16813620C22semidirect
32. #16813609C22semidirect
33. #16813608C22semidirect
38. #16813602C22semidirect
40. #16813601C22semidirect
43. #16813599C22semidirect
44. #16813599C22semidirect
45. #16813598C22semidirect
51. Q8×Dic113C22semidirect
52. C4×Dic226C22semidirect
53. #16813587C22semidirect
61. C2×#41×C113C22semidirect
63. #124×C113C22semidirect
64. C22⋊Q8×C113C22semidirect
66. C4×Q8×C113C22semidirect
83. Q8×D113D4semidirect
178. C22×C1132−1+4semidirect
189. C2×D1132−1+4semidirect
209. C113#3486semidirect
219. #548D113semidirect
229. C2×#41C2×D113semidirect
245. #124C2×D113semidirect
258. C22⋊Q8C2×D113semidirect
268. C4×Q8C2×D113semidirect
397. C22#16813669semidirect

Maximal subgroups

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

subgroupindex
C2×#168136692
C22×Q8×D1132
Q8×D4522
D4×Dic2262
Q8×C22⋊D1132
#202992612
#202992562
#202992552
#202992542
#202991712
#202991682
#202991082
#202991072
#202990742
#548×C1132
#3486113