Finite Group Databasebeta

← #29697694

Subgroups — #29697694

Order 15616.

838 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
1C22×#14519708C211yesmaximal, characteristic
2#21448686C211yesmaximal, characteristic
3#21446559C212yesmaximal, complemented
4#21446404C212yesmaximal, complemented
5#21446394C212yesmaximal, complemented
6#21445909C211yesmaximal, characteristic
7#21445903C211yesmaximal, characteristic
8#21445846C211yesmaximal, characteristic
9#21445717C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic
11C61×#2345C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519733—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15C2×#14519710C2212yes—
16C2×#14519708C418yescomplemented
17C2×#14519708C2211yescharacteristic
18C2×#14519708C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×C22⋊#92418C2211yescharacteristic
22C2×#14519688C2212yes—
23#16839660C2212yes—
24#16839647—22no—
25#16839623—21no—
26#16839622C2211yescharacteristic
27C4×C4⋊Dic61C2212yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#389C2212yes—
36C61×#387—22no—
37C61×#380—21no—
38C61×#380C2211yescharacteristic
39C4×C4⋊C4×C61C2212yes—
40C4×C22⋊C4×C61—21no—
41C4×C22⋊C4×C61C2211yescharacteristic
42C23×Dic61—21no—
43C22×#92418—24no—
44C22×#92418C2×C414yes—
45C22×#92418C2312yes—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48#14519733—44no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61C2×C414yes—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—22no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—22no—
58#14519710—24no—
59#14519708C2×C414yes—
60#14519708C2×C414yes—
61#14519708C2×C414yes—
62#14519708C2×C414yes—
63C22⋊#92418—22no—
64#14519688—24no—
65#14519684—22no—
66#14519684—21no—
67#14519684—22no—
68C23×C4×C61—21no—
69C23×C4×C61C2311yescharacteristic
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61C2×C414yes—
74C2×C4⋊C4×C61C2312yes—
75C2×C4⋊C4×C61C2311yescharacteristic
76C2×C4⋊C4×C61C2311yescharacteristic
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61D411yescharacteristic
79C2×C22⋊C4×C61D411yescharacteristic
80C2×C42×C61—22no—
81C2×C42×C61C2312yes—
82#95×C61—22no—
83#95×C61—22no—
84#95×C61—21no—
85#95×C61D412yes—
86C24×C61C2×D411yescharacteristic
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—24no—
92C22×C4×C61—22no—
93C22×C4×C61—82no—
94C22×C4×C61C22×C412yes—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4112yes—
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—24no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61—42no—
105C22×Dic61—42no—
106C4⋊C4×C61—24no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—82no—
112C4⋊C4×C61C22×C412yes—
113C4⋊C4×C61C22×C412yes—
114C22⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C22⋊C4×C61—42no—
117C42×C61—22no—
118C42×C61—22no—
119C42×C61—82no—
120C22⋊Dic61—42no—
121C4⋊Dic61—24no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—22no—
126C4⋊Dic61C22×C412yes—
127C4⋊Dic61C22×C412yes—
128#11758196—44no—
129C4×Dic61—44no—
130C2×#92418—22no—
131C2×#92418—28no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418C22×C412yes—
136C2×#92418C22×C412yes—
137C2×#92418#4114yes—
138C23×C61—22no—
139C23×C61—22no—
140C23×C61C4×D412yes—
141C23×C61AGL(1,ℤ/8)12yes—
142C23×C61#13712yes—
143C23×C61C22×D411yescharacteristic
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—84no—
162C2×C4×C61C4×D412yes—
163C2×C4×C61C23×C411yescharacteristic
164C2×C4×C61C2×#4112yes—
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C612−1+412yes—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—48no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177#92418—22no—
178#92418—42no—
179#92418#11714yescomplemented
180#22615—611nomaximal, Sylow
181C22×C61—24no—
182C22×C61—24no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61#44014yes—
186C22×C61C2×C4×D411yescharacteristic
187C22×C61#54712yes—
188C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
189C22×C61C2×#13711yescharacteristic
190C22×C61#57712yes—
191C4×C61—22no—
192C4×C61—42no—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61—82no—
197C4×C61C2×#11711yescharacteristic
198C4×C61#51911yescharacteristic
199C4×C61#55212yes—
200Dic61—42no—
201Dic61—42no—
202Dic61—82no—
203C22×#106—611no—
204#2345D6111yescharacteristic
205#2344—611no—
206#1944—612no—
207#1920—612no—
208#1919—612no—
209#1824—611no—
210#1823—611no—
211#1808—611no—
212#1776—612no—
213#1752—611no—
214C2×C61—24no—
215C2×C61#234411yescharacteristic
216C2×C61C2×#44011yescharacteristic
217C2×C61#294911yescharacteristic
218C2×C61#334412yes—
219C2×C61#334512yes—
220C23×C8—611no—
221C2×#117—1221no—
222C22×C4⋊C4—611no—
223C22×C4⋊C4C2×D6111yescharacteristic
224C2×#107—612no—
225C2×#106—618no—
226C2×#106—611no—
227C2×#106—611no—
228C2×#106—611no—
229C2×#106—611no—
230C2×C22⋊C8—611no—
231C2×#97—612no—
232#389—612no—
233#389C2×D6112yes—
234#387—22no—
235#387—1222no—
236#381—1221no—
237#380—21no—
238#380—611no—
239#380C2×D6111yescharacteristic
240C4×C4⋊C4—612no—
241C4×C4⋊C4C2×D6112yes—
242C4×C22⋊C4—21no—
243C4×C22⋊C4C2×D6111yescharacteristic
244#340—612no—
245#340—612no—
246#340—612no—
247#340—612no—
248#340—612no—
249#336—612no—
250C61#2261511yescharacteristic, complemented, Sylow
251C23×C4—21no—
252C23×C4—1221no—
253C23×C4C22×D6111yescharacteristic
254C22×C8—611no—
255C22×C8—614no—
256C22×C8—611no—
257C22×C8—612no—
258C22×C8—1224no—
259#117—2444no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—614no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—612no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1222no—
270C2×C4⋊C4C4×D6114yes—
271C2×C4⋊C4C22×D6112yes—
272C2×C4⋊C4C22×D6111yescharacteristic
273C2×C4⋊C4C22×D6111yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4C22⋊D6111yescharacteristic
277C2×C22⋊C4C22⋊D6111yescharacteristic
278C2×C42—22no—
279C2×C42—1222no—
280C2×C42C22×D6112yes—
281#107—1224no—
282#106—614no—
283#106—614no—
284#106—614no—
285#106—614no—
286C22⋊C8—1222no—
287#97—1224no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—1222no—
292#95—1221no—
293#95—1222no—
294#95C22⋊D6112yes—
295C24C2×C22⋊D6111yescharacteristic
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—82no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1224no—
306C22×C4—1222no—
307C22×C4—2442no—
308C22×C4—2442no—
309C22×C4C2×C4×D6112yes—
310C22×C4#1175822212yes—
311C22×C4C2×C22⋊D6112yes—
312C22×C4C2×C22⋊D6112yes—
313C22×C4C2×C22⋊D6111yescharacteristic
314C22×C4C23×D6111yescharacteristic
315C2×C8—612no—
316C2×C8—614no—
317C2×C8—612no—
318C2×C8—1228no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8—1222no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—82no—
329C4⋊C4—612no—
330C4⋊C4—612no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1224no—
334C4⋊C4—1222no—
335C4⋊C4—1222no—
336C4⋊C4—2444no—
337C4⋊C4C2×C4×D6112yes—
338C4⋊C4C2×C4×D6112yes—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—42no—
342C22⋊C4—2442no—
343C42—22no—
344C42—22no—
345C42—82no—
346C42—2444no—
347C23—22no—
348C23—22no—
349C23C4×C22⋊D6112yes—
350C23#1451980212yes—
351C23#1451980812yes—
352C23C22×C22⋊D6111yescharacteristic
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2448no—
381C2×C4C4×C22⋊D6112yes—
382C2×C4C22×C4×D6111yescharacteristic
383C2×C4#41×D6112yes—
384C2×C4C2×#1175822211yescharacteristic
385C2×C4#1451982412yes—
386C8—614no—
387C8—1222no—
388C8—2442no—
389C22—22no—
390C22—24no—
391C22—24no—
392C22—22no—
393C22#1683986014yes—
394C22C2×C4×C22⋊D6111yescharacteristic
395C22#1684041112yes—
396C22C2×#1451980211yescharacteristic
397C22C2×#1451980811yescharacteristic
398C22#1684057312yes—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—4882no—
406C4—2442no—
407C4—2442no—
408C4#117×D6111yescharacteristic
409C4#1684023911yescharacteristic
410C4#1684044512yes—
411C2—24no—
412C2#2144868911yescharacteristic
413C2C2×#1683986011yescharacteristic
414C2#2145467611yescharacteristic
415C2#2145826812yes—
416C2#2145828312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2345C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#2345D61
hypocenterC61#22615
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #21446559C2semidirect
4. #21446404C2semidirect
5. #21446394C2semidirect
9. #21445717C2semidirect
16. C2×#14519708C4semidirect
179. #92418#117semidirect
250. C61#22615semidirect

Maximal subgroups

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

subgroupindex
C22×#145197082
#214486862
#214465592
#214464042
#214463942
#214459092
#214459032
#214458462
#214457172
#214455972
C61×#23452
#2261561