Finite Group Databasebeta

← #29680523

Subgroups — #29680523

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839824C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic
3#21448158C211yesmaximal, characteristic
4#21445972C212yesmaximal, complemented
5#21445934C212yesmaximal, complemented
6#21445802C212yesmaximal
7#21445796C212yesmaximal
8#21445676C211yesmaximal, characteristic, complemented
9#21445673C211yesmaximal, characteristic, complemented
10C4×#14519706C211yesmaximal, characteristic, complemented
11C61×#1768C211yesmaximal, characteristic, complemented
12C2×#14519734C2211yescharacteristic
13C2×#14519734C2211yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839824—24no—
16#16839821—24no—
17C2×C4⋊#92418C2212yes—
18C2×#14519706C2212yes—
19C2×#14519706C2212yes—
20C2×#14519706C2212yes—
21C2×#14519706C2212yes—
22C2×#14519689C2212yes—
23C2×#14519689C2212yes—
24C2×#14519688C2211yescharacteristic
25C2×C4×#92418C2211yescharacteristic
26#16839613C412yes—
27#16839613C412yes—
28#16839609C412yes—
29#16839609C412yes—
30#16839608C2211yescharacteristic
31C42×Dic61C2211yescharacteristic
32C42×Dic61C2211yescharacteristic
33#16839517C2212yes—
34#16839516C2212yes—
35#16839516C2212yes—
36C2×C4⋊C8×C61C2212yes—
37C2×#97×C61C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#376C2211yescharacteristic
40C61×#336C2212yes—
41C22×C4×D61C2×C412yes—
42C22×C4×D61C2311yescharacteristic
43C22×#92418C2312yes—
44C22×#92418C2312yes—
45#14519734—24no—
46#14519734D414yes—
47#14519734Q814yes—
48C2×#11758196C2×C411yescharacteristic
49C2×#11758196C2×C411yescharacteristic
50C2×#11758196C2×C412yes—
51C2×#11758198C2×C411yescharacteristic
52C2×#11758198C2×C411yescharacteristic
53C2×#11758198C2×C412yes—
54C2×C4×Dic61C2×C412yes—
55C2×C4×Dic61C2×C412yes—
56C2×C4×Dic61C2×C412yes—
57C2×C4×Dic61C2×C412yes—
58C2×C4×Dic61C2×C412yes—
59C2×C4×Dic61C2311yescharacteristic
60C2×C4×Dic61C2311yescharacteristic
61C2×C4×Dic61C2311yescharacteristic
62C42×D61—24no—
63C4⋊#92418—24no—
64#14519706—24no—
65#14519706—24no—
66#14519706—24no—
67#14519706—24no—
68#14519689—24no—
69#14519689—24no—
70#14519688—22no—
71C4×#92418—22no—
72#14519684C2×C412yes—
73#14519684C2×C412yes—
74C22×C8×C61C2312yes—
75C22×C8×C61C2312yes—
76C2×C42×C61C2311yescharacteristic
77C2×C42×C61C2311yescharacteristic
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—24no—
80#97×C61—22no—
81C4×C8×C61—22no—
82#95×C61C2×C412yes—
83#95×C61C2×C412yes—
84C23×D61C22×C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C22×C412yes—
88C22×C4×C61C22×C412yes—
89C22×C4×C61C2411yescharacteristic
90C22×Dic61C22×C411yescharacteristic
91C22×Dic61C22×C412yes—
92C22×Dic61C22×C411yescharacteristic
93C22×Dic61C22×C411yescharacteristic
94C22×Dic61C22×C412yes—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—24no—
98C2×C4×D61—24no—
99C2×C4×D61—42no—
100C2×C4×D61C2×D412yes—
101C2×C4×D61C2×Q812yes—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C42×C61—22no—
109C42×C61C2×D412yes—
110C42×C61C2×Q812yes—
111C42×C61#4114yes—
112#11758198—22no—
113#11758198—22no—
114#11758198C4⋊C418yes—
115#11758196—22no—
116#11758196—22no—
117#11758196C4⋊C418yes—
118C4×Dic61—24no—
119C4×Dic61—22no—
120C4×Dic61—24no—
121C4×Dic61—24no—
122C4×Dic61—42no—
123C4×Dic61—42no—
124C4×Dic61C2×D412yes—
125C4×Dic61C2×Q812yes—
126C4×Dic61#4114yes—
127C2×#92418—22no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—22no—
133C23×C61C23×C411yescharacteristic
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—22no—
137C22×D61—22no—
138C22×D61C2×C4⋊C412yes—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61C2×C4⋊C414yes—
148C2×C4×C61#11714yes—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C22×Q811yescharacteristic
151C2×C4×C61C2×#4112yes—
152C2×C4×C61C2×#4112yes—
153C2×Dic61—22no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61C2×C4⋊C414yes—
162C2×Dic61C2×C4⋊C412yes—
163C2×Dic61#11712yes—
164C2×Dic61#11712yes—
165C2×Dic61#13114yes—
166C4×D61—24no—
167C4×D61—24no—
168C4×D61—48no—
169C8×C61—42no—
170C8×C61—42no—
171#92418—42no—
172#92418—42no—
173#20813—611nomaximal, Sylow
174C22×C61C22×C4⋊C411yescharacteristic
175C22×C61C2×#11711yescharacteristic
176C22×C61C2×#11711yescharacteristic
177C22×C61C2×#13112yes—
178C22×C61#56812yes—
179C2×D61—24no—
180C2×D61—22no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—42no—
184C4×C61—24no—
185C4×C61—24no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61#53212yes—
189C4×C61#53312yes—
190Dic61—24no—
191Dic61—22no—
192Dic61—22no—
193Dic61—24no—
194Dic61—22no—
195C2×#433—611no—
196C2×#433—611no—
197C4×#117—611no—
198#1843—612no—
199#1831—612no—
200#1798—612no—
201#1797—612no—
202#1768—611no—
203#1768D6111yescharacteristic, complemented
204#1767—611no—
205C4×C4⋊C8—611no—
206C2×C61#228511yescharacteristic
207C2×C61#290612yes—
208C2×C61#292211yescharacteristic
209C2×C61#292311yescharacteristic
210C2×C61#292412yes—
211D61—42no—
212C2×#117—611no—
213C2×#117—611no—
214C22×C42—611no—
215#433—1224no—
216#433—1224no—
217C2×C4⋊C8—612no—
218C2×C4⋊C8—612no—
219C2×C4⋊C8—612no—
220C2×C4⋊C8—612no—
221C2×C4⋊C8—612no—
222C2×C4⋊C8C2×D6112yes—
223C2×C22⋊C8—612no—
224C2×C22⋊C8—612no—
225C2×#97—611no—
226C2×#97C2×D6111yescharacteristic
227C2×C4×C8—611no—
228C2×C4×C8C2×D6111yescharacteristic
229C4×C4⋊C4—612no—
230C4×C4⋊C4—612no—
231C4×C22⋊C4—612no—
232C4×C22⋊C4—612no—
233#376—611no—
234#376C2×D6111yescharacteristic
235C43—611no—
236C43—611no—
237#336—612no—
238#336—612no—
239#336—612no—
240#336C2×D6112yes—
241C61#2081311yescharacteristic, complemented, Sylow
242C23×C4—611no—
243C23×C4—612no—
244C22×C8—612no—
245C22×C8—612no—
246C22×C8C22×D6112yes—
247C22×C8C22×D6112yes—
248#117—614no—
249#117—614no—
250#117—1224no—
251C2×C4⋊C4—612no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C22⋊C4—612no—
255C2×C22⋊C4—611no—
256C2×C22⋊C4—611no—
257C2×C42—611no—
258C2×C42—611no—
259C2×C42—612no—
260C2×C42—612no—
261C2×C42—612no—
262C2×C42—611no—
263C2×C42—612no—
264C2×C42—612no—
265C2×C42—1224no—
266C2×C42C22×D6111yescharacteristic
267C2×C42C22×D6111yescharacteristic
268C2×C42C22×D6111yescharacteristic
269C4⋊C8—24no—
270C4⋊C8—1224no—
271C4⋊C8—1224no—
272C4⋊C8—1224no—
273C4⋊C8—1224no—
274C4⋊C8—1224no—
275C22⋊C8—1224no—
276C22⋊C8—1224no—
277#97—22no—
278#97—1222no—
279C4×C8—22no—
280C4×C8—1222no—
281#95—612no—
282#95—612no—
283#95C4×D6112yes—
284#95C4×D6112yes—
285C24—611no—
286C22×C4—612no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—612no—
290C22×C4—611no—
291C22×C4—612no—
292C22×C4—612no—
293C22×C4—1224no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1224no—
297C22×C4—2442no—
298C22×C4C2×C4×D6111yescharacteristic
299C22×C4C2×C4×D6111yescharacteristic
300C22×C4C2×C4×D6112yes—
301C22×C4C2×C4×D6112yes—
302C22×C4C23×D6111yescharacteristic
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—1222no—
310C2×C8—1222no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C4⋊C4—618no—
316C4⋊C4—1222no—
317C4⋊C4—1222no—
318C22⋊C4—618no—
319C22⋊C4—1222no—
320C22⋊C4—1222no—
321C42—22no—
322C42—612no—
323C42—612no—
324C42—614no—
325C42—1224no—
326C42—1224no—
327C42—1222no—
328C42—1224no—
329C42—2442no—
330C42—2442no—
331C42#1175822214yes—
332C42D4×D6112yes—
333C42Q8×D6112yes—
334C23—612no—
335C23—1221no—
336C23—1221no—
337C23—1222no—
338C23—1222no—
339C23C22×C4×D6111yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—614no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—614no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—1224no—
362C2×C4—1224no—
363C2×C4—2448no—
364C2×C4C4⋊C4×D6114yes—
365C2×C4#1451973414yes—
366C2×C4C2×D4×D6111yescharacteristic
367C2×C4C2×Q8×D6111yescharacteristic
368C2×C4#41×D6112yes—
369C2×C4C2×#1175822212yes—
370C8—42no—
371C8—42no—
372C8—2442no—
373C8—2442no—
374C22—1222no—
375C22—1224no—
376C22—2441no—
377C22—2441no—
378C22—2442no—
379C22C2×C4⋊C4×D6111yescharacteristic
380C22#117×D6111yescharacteristic
381C22C2×#1451973411yescharacteristic
382C22#131×D6112yes—
383C22#1684052712yes—
384C4—24no—
385C4—24no—
386C4—41no—
387C4—41no—
388C4—1222no—
389C4—1224no—
390C4—1222no—
391C4—1222no—
392C4—1224no—
393C4#1684030912yes—
394C4#1684031612yes—
395C2—2442no—
396C2#2144822311yescharacteristic
397C2#2145438612yes—
398C2#2145446011yescharacteristic
399C2#2145446711yescharacteristic
400C2#2145447412yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1768C2
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1768D61
hypocenterC61#20813
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1768

Complements

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

NH ≅ G/Ntype
4. #21445972C2semidirect
5. #21445934C2semidirect
8. #21445676C2semidirect
9. #21445673C2semidirect
10. C4×#14519706C2semidirect
11. C61×#1768C2semidirect
203. #1768D61semidirect
241. C61#20813semidirect

Maximal subgroups

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

subgroupindex
C2×#168398242
C2×#168398212
#214481582
#214459722
#214459342
#214458022
#214457962
#214456762
#214456732
C4×#145197062
C61×#17682
#2081361