Finite Group Databasebeta

← #29680433

Subgroups — #29680433

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839799C212yesmaximal
2C2×#16839797C212yesmaximal
3C2×#97×D61C211yesmaximal, characteristic
4#21448225C211yesmaximal, characteristic
5#21448224C211yesmaximal, characteristic
6#21445988C211yesmaximal, characteristic, complemented
7#21445987C211yesmaximal, characteristic, complemented
8#21445986C212yesmaximal, complemented
9#21445985C211yesmaximal, characteristic, complemented
10#21445984C212yesmaximal, complemented
11C61×#1847C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519735C2212yes—
14C2×#14519735C2211yescharacteristic
15C2×#14519735C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839799—28no—
18#16839797—28no—
19C2×#14519711C2212yes—
20C2×#14519710C2212yes—
21C2×#14519710C2212yes—
22C2×#14519709C2212yes—
23C2×#14519708C2212yes—
24C2×#14519708C2212yes—
25C2×#14519688C2211yescharacteristic
26C2×#14519687C2212yes—
27#97×D61C418yescomplemented
28#16839632C2211yescharacteristic
29#16839631C2211yescharacteristic
30#16839631C2211yescharacteristic
31#16839628C2211yescharacteristic
32#16839627C2212yes—
33#16839621—22no—
34#16839619—22no—
35C4⋊C4×Dic61—22no—
36#16839613—22no—
37C2×#107×C61C2212yes—
38C2×#106×C61C2212yes—
39C2×#97×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C22×C4×D61D412yes—
43C22×C4×D61C2311yescharacteristic
44C22×#92418C2312yes—
45C2×C8×D61C2×C418yes—
46C2×C8×D61D418yes—
47C2×C8×D61Q818yes—
48#14519735—24no—
49#14519735—24no—
50#14519735—24no—
51#14519735—24no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758198—22no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61D412yes—
65C2×C4×Dic61C2311yescharacteristic
66C42×D61C2×C414yes—
67C42×D61Q814yes—
68#14519711—24no—
69#14519710—24no—
70#14519710—24no—
71#14519709—24no—
72#14519708—24no—
73#14519708—24no—
74#14519688C2×C414yes—
75#14519687C2×C418yes—
76#14519684—22no—
77#14519684—24no—
78C22×C8×C61C2312yes—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61C2311yescharacteristic
81C2×C4⋊C4×C61C2311yescharacteristic
82C2×C4⋊C4×C61C2312yes—
83C2×C42×C61C2311yescharacteristic
84#107×C61—24no—
85#106×C61—24no—
86#97×C61C2×C414yes—
87#95×C61—22no—
88C23×D61C2×D411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2411yescharacteristic
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—22no—
97C22×Dic61C2×D412yes—
98C22×Dic61C2×D411yescharacteristic
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×C4×D61C4⋊C418yes—
102C2×C4×D61C22×C412yes—
103C2×C4×D61C2×D414yes—
104C2×C4×D61C2×Q814yes—
105C2×C4×D61C2×Q812yes—
106C2×C8×C61C22×C414yes—
107C2×C8×C61C2×D414yes—
108C2×C8×C61C2×Q814yes—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—44no—
114C42×C61C22×C412yes—
115C42×C61C2×Q812yes—
116#11758198—44no—
117#11758198—42no—
118#11758198—42no—
119C4⋊Dic61—22no—
120C4⋊Dic61—22no—
121C4⋊Dic61—22no—
122C4⋊Dic61—22no—
123C4⋊Dic61—44no—
124#11758196—44no—
125#11758196—44no—
126C4×Dic61—42no—
127C4×Dic61—42no—
128C4×Dic61—44no—
129C4×Dic61C4⋊C418yes—
130C4×Dic61C22×C412yes—
131C4×Dic61C2×Q812yes—
132C2×#92418C22×C414yes—
133C2×#92418C2×D414yes—
134C2×#92418C2×Q814yes—
135C8×D61C4⋊C4132yescomplemented
136C23×C61C22×D411yescharacteristic
137C22×D61—22no—
138C22×D61—22no—
139C22×D61C2×C4⋊C412yes—
140C22×D61#12712yes—
141C22×D61C4⋊Q812yes—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—44no—
150C2×C4×C61C2×C4⋊C414yes—
151C2×C4×C61C23×C411yescharacteristic
152C2×C4×C61C22×D412yes—
153C2×C4×C61C22×Q812yes—
154C2×C4×C61C22×Q811yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—44no—
163C2×Dic61C2×C4⋊C414yes—
164C2×Dic61C2×C4⋊C412yes—
165C2×Dic61#12712yes—
166C2×Dic61C4⋊Q812yes—
167C4×D61—48no—
168C4×D61C2×C4⋊C418yes—
169C4×D61C4⋊Q818yes—
170C8×C61C2×C4⋊C418yes—
171#92418C2×C4⋊C418yes—
172#20799—611nomaximal, Sylow
173C22×C61C22×C4⋊C411yescharacteristic
174C22×C61C2×#12711yescharacteristic
175C22×C61C2×C4⋊Q811yescharacteristic
176C22×C61#57512yes—
177C22×C61#57512yes—
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—24no—
181C2×D61—24no—
182C2×D61#38414yes—
183C4×C61—24no—
184C4×C61—81no—
185C4×C61—82no—
186C4×C61—81no—
187C4×C61C22×C4⋊C412yes—
188C4×C61C2×C4⋊Q812yes—
189Dic61—24no—
190Dic61—81no—
191Dic61—81no—
192Dic61—82no—
193Dic61#38414yes—
194C2×#427—612no—
195C2×#427—612no—
196C22×#97—611no—
197#2285—611no—
198#2285—611no—
199#1847—612no—
200#1847—612no—
201#1847—611no—
202#1847—611no—
203#1847—611no—
204#1847D6111yescharacteristic, complemented
205C2×C61C2×#38411yescharacteristic
206C2×C61#291112yes—
207C2×C61#315011yescharacteristic
208C2×C61#315011yescharacteristic
209C2×C61#316212yes—
210D61—24no—
211C23×C8—612no—
212C2×#117—612no—
213C2×#117—611no—
214C2×#117—611no—
215C22×C42—611no—
216#427—1228no—
217#427—1228no—
218C2×#107—612no—
219C2×#107—612no—
220C2×#107—612no—
221C2×#107C2×D6112yes—
222C2×#106—612no—
223C2×#106—612no—
224C2×#106—612no—
225C2×#106C2×D6112yes—
226C2×#97—612no—
227C2×#97—611no—
228C2×#97—618no—
229C2×#97C2×D6111yescharacteristic
230#384—611no—
231#384—611no—
232#384—611no—
233#384C2×D6111yescharacteristic
234#382—612no—
235#382—611no—
236#382C2×D6111yescharacteristic
237#380—1222no—
238#380—1222no—
239C4×C4⋊C4—1222no—
240C4×C4⋊C4—1222no—
241C61#2079911yescharacteristic, complemented, Sylow
242C23×C4—611no—
243C23×C4—612no—
244C22×C8—612no—
245C22×C8—618no—
246C22×C8—618no—
247C22×C8—618no—
248C22×C8C22×D6112yes—
249#117—1224no—
250#117—1224no—
251#117—1224no—
252#117—1224no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—612no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1222no—
259C2×C4⋊C4—1222no—
260C2×C4⋊C4C22×D6112yes—
261C2×C4⋊C4C22×D6111yescharacteristic
262C2×C4⋊C4C22×D6111yescharacteristic
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1222no—
266C2×C42—612no—
267C2×C42—611no—
268C2×C42—614no—
269C2×C42—614no—
270C2×C42—1221no—
271C2×C42—1221no—
272C2×C42—1222no—
273C2×C42C22×D6111yescharacteristic
274#107—24no—
275#107—1224no—
276#107—1224no—
277#107—1224no—
278#106—24no—
279#106—1224no—
280#106—1224no—
281#106—1224no—
282#97—618no—
283#97—614no—
284#97C4×D6114yes—
285#95—22no—
286#95—1224no—
287#95—1222no—
288C24—611no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—611no—
293C22×C4—612no—
294C22×C4—614no—
295C22×C4—612no—
296C22×C4—614no—
297C22×C4—612no—
298C22×C4—618no—
299C22×C4—1221no—
300C22×C4—1222no—
301C22×C4—1221no—
302C22×C4—2442no—
303C22×C4—2442no—
304C22×C4D4×D6112yes—
305C22×C4C23×D6111yescharacteristic
306C2×C8—614no—
307C2×C8—614no—
308C2×C8—614no—
309C2×C8—6132no—
310C2×C8C2×C4×D6114yes—
311C2×C8D4×D6114yes—
312C2×C8Q8×D6114yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—1222no—
319C4⋊C4—1222no—
320C4⋊C4—1222no—
321C4⋊C4—1222no—
322C4⋊C4—2444no—
323C4⋊C4—2444no—
324C4⋊C4—2444no—
325C22⋊C4—2444no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—612no—
329C42—612no—
330C42—618no—
331C42—2444no—
332C42—2442no—
333C42—2442no—
334C42C2×C4×D6112yes—
335C42Q8×D6112yes—
336C23—612no—
337C23—612no—
338C23—612no—
339C23—1222no—
340C23—1222no—
341C23C2×D4×D6111yescharacteristic
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—618no—
354C2×C4—618no—
355C2×C4—614no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—2448no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2444no—
365C2×C4C4⋊C4×D6114yes—
366C2×C4C22×C4×D6111yescharacteristic
367C2×C4C2×D4×D6112yes—
368C2×C4C2×Q8×D6112yes—
369C2×C4C2×Q8×D6111yescharacteristic
370C8—618no—
371C8C4⋊C4×D6118yes—
372C22—614no—
373C22—1224no—
374C22—1222no—
375C22—1222no—
376C22—1224no—
377C22C2×C4⋊C4×D6111yescharacteristic
378C22#127×D6111yescharacteristic
379C22C4⋊Q8×D6111yescharacteristic
380C22#1684055812yes—
381C22#1684056112yes—
382C4—24no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4—614no—
387C4—1224no—
388C4—4881no—
389C4—4881no—
390C4—4882no—
391C4C2×C4⋊C4×D6112yes—
392C4C4⋊Q8×D6112yes—
393C2—1224no—
394C2#384×D6111yescharacteristic
395C2#2145440612yes—
396C2#2145626111yescharacteristic
397C2#2145626311yescharacteristic
398C2#2145634012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#20799
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
6. #21445988C2semidirect
7. #21445987C2semidirect
8. #21445986C2semidirect
9. #21445985C2semidirect
10. #21445984C2semidirect
11. C61×#1847C2semidirect
27. #97×D61C4semidirect
135. C8×D61C4⋊C4semidirect
204. #1847D61semidirect
241. C61#20799semidirect

Maximal subgroups

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

subgroupindex
C2×#168397992
C2×#168397972
C2×#97×D612
#214482252
#214482242
#214459882
#214459872
#214459862
#214459852
#214459842
C61×#18472
#2079961