Finite Group Databasebeta

← #29689679

Subgroups — #29689679

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459208C211yesmaximal, characteristic
2#21454381C211yesmaximal, characteristic, complemented
3#21448316C211yesmaximal, characteristic, complemented
4#21446799C211yesmaximal, characteristic, complemented
5#21446796C211yesmaximal, characteristic, complemented
6#21446652C211yesmaximal, characteristic, complemented
7#21446118C214yesmaximal, complemented
8#21445837C214yesmaximal, complemented
9C61×#1962C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840381C414yescomplemented
12#16840373C412yescomplemented
13#16840366C412yescomplemented
14#16840357—22no—
15#16840349—22no—
16#16840319C2211yescharacteristic
17#16840292—24no—
18#16840269C2212yes—
19#41×Dic61C2212yes—
20C2×#14519734C2211yescharacteristic
21#16839787—24no—
22#16839786C2211yescharacteristic
23#16839785C2212yes—
24C2×#14519704C2214yes—
25#16839653—22no—
26#16839646—22no—
27C4×#11758196C2212yes—
28C4×#11758198C2212yes—
29#16839608C2211yescharacteristic
30#16839523C2214yes—
31#16839522C2214yes—
32C61×#532C2211yescharacteristic
33C61×#423C2211yescharacteristic
34C2×#104×C61C2214yes—
35C43×C61C2211yescharacteristic
36C22×C4×D61C2311yescharacteristic
37C2×#11758194C2312yes—
38C2×#11758189—22no—
39M16×D61—44no—
40C2×C8×D61—22no—
41#14519777C2×C412yes—
42#14519772—24no—
43#14519770—24no—
44#14519769C2×C412yes—
45#14519767—24no—
46#14519763—24no—
47#14519761—24no—
48#14519758—22no—
49#14519758—24no—
50#14519756—22no—
51#14519754—24no—
52#14519753C2×C414yes—
53#14519749—24no—
54#14519748C2×C414yes—
55#14519745—22no—
56Q8×Dic61C2×C414yes—
57C4×Dic122—22no—
58D4×Dic61—24no—
59D4×Dic61C2×C414yes—
60C4×C22⋊D61—24no—
61#14519736—22no—
62#14519736C2312yes—
63#14519734C2×C412yes—
64#14519734C2×C412yes—
65#14519734D414yes—
66C2×#11758196—22no—
67C2×#11758196D412yes—
68C2×#11758198—22no—
69C2×#11758198D412yes—
70C2×C4×Dic61—22no—
71C2×C4×Dic61C2312yes—
72C2×C4×Dic61C2311yescharacteristic
73C4⋊#92418—22no—
74#14519706—24no—
75#14519704—44no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79C2×#41×C61C2312yes—
80C2×M16×C61C2312yes—
81C4⋊Q8×C61C2×C411yescharacteristic
82#127×C61C2×C411yescharacteristic
83#124×C61C2×C412yes—
84#121×C61—22no—
85C4×D4×C61—22no—
86C2×C42×C61—22no—
87C2×C42×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—22no—
90#104×C61—44no—
91C23×D61C2×D411yescharacteristic
92#41×C61—24no—
93#41×C61—24no—
94C2×Q8×C61C22×C412yes—
95C2×D4×C61—22no—
96C2×D4×C61—22no—
97C2×D4×C61C22×C412yes—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2411yescharacteristic
102C2×C22⋊D61—24no—
103C22×Dic61—24no—
104C22×Dic61—22no—
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D412yes—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61C22×C412yes—
110C2×C4×D61C2×D412yes—
111C2×Dic122—22no—
112M16×C61—24no—
113C2×C8×C61—22no—
114C4⋊C4×C61—22no—
115C22⋊C4×C61—24no—
116C42×C61—22no—
117C42×C61—22no—
118C42×C61—44no—
119C42×C61C22×C411yescharacteristic
120C42×C61C22×C411yescharacteristic
121C42×C61C2×D412yes—
122C22⋊Dic61—22no—
123C22⋊Dic61—24no—
124C22⋊Dic61—24no—
125#11758198—24no—
126#11758198—44no—
127#11758198C22⋊C414yes—
128#11758198C22⋊C414yes—
129C4⋊Dic61—22no—
130C4⋊Dic61—24no—
131C4⋊Dic61C22×C414yes—
132#11758196—24no—
133#11758196—24no—
134#11758196—44no—
135#11758196C22⋊C414yes—
136#11758196C22⋊C414yes—
137C4×Dic61—22no—
138C4×Dic61—24no—
139C4×Dic61—22no—
140C4×Dic61—22no—
141C4×Dic61C22×C412yes—
142C4×Dic61C2×D412yes—
143#11758194—24no—
144C2×#92418—22no—
145#11758189—44no—
146C8×D61—44no—
147C23×C61—22no—
148C23×C61C22×D411yescharacteristic
149C22×D61—22no—
150C22×D61—42no—
151C22×D61C2×C22⋊C412yes—
152Q8×C61—24no—
153D4×C61—24no—
154D4×C61—44no—
155D4×C61—82no—
156C2×C4×C61—24no—
157C2×C4×C61—22no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61C2×C22⋊C412yes—
164C2×C4×C61C2×C22⋊C412yes—
165C2×C4×C61C23×C411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61H(5,𝔽2)12yes—
168C2×C4×C612−1+412yes—
169C22⋊D61—84no—
170C2×Dic61—22no—
171C2×Dic61—24no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—84no—
177C2×Dic61C2×C22⋊C414yes—
178C2×Dic61C2×C22⋊C412yes—
179C4×D61—24no—
180C4×D61—22no—
181C4×D61—22no—
182Dic122—82no—
183C8×C61—42no—
184#92418—42no—
185#21858—611nomaximal, Sylow
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—44no—
189C22×C61C22×C22⋊C411yescharacteristic
190C22×C61#53712yes—
191C2×D61—22no—
192C2×D61—24no—
193C2×D61—82no—
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—44no—
197C4×C61—42no—
198C4×C61#52012yes—
199C4×C61#53511yescharacteristic
200C4×C61#53611yescharacteristic
201Dic61—22no—
202Dic61—24no—
203Dic61—44no—
204Dic61—82no—
205#3455—611no—
206#2905—611no—
207#2298—611no—
208#1995—611no—
209#1994—611no—
210#1963—611no—
211#1962D6111yescharacteristic, complemented
212#1871—614no—
213#1806—614no—
214C2×C61—22no—
215C2×C61—82no—
216C2×C61#231111yescharacteristic
217C2×C61#290112yes—
218D61—42no—
219C22×M16—612no—
220#542—614no—
221#541—612no—
222#540—612no—
223#539—1222no—
224#538—1222no—
225#533—611no—
226#532C2×D6111yescharacteristic
227#529—1224no—
228#525—612no—
229C4×#41—612no—
230C2×#117—611no—
231#424—1224no—
232#423—612no—
233#423—611no—
234#423C2×D6111yescharacteristic
235C2×#104—614no—
236C2×#104C2×D6114yes—
237#388—1222no—
238#387—1222no—
239C4×C4⋊C4—612no—
240C4×C22⋊C4—612no—
241#376—611no—
242C43C2×D6111yescharacteristic
243#339—614no—
244#339—614no—
245C61#2185811yescharacteristic, complemented, Sylow
246C2×#41C22×D6112yes—
247C23×C4—611no—
248C2×M16—612no—
249C2×M16—1222no—
250C2×M16—2444no—
251C2×M16C22×D6112yes—
252C22×C8—1222no—
253C4⋊Q8—612no—
254C4⋊Q8C4×D6111yescharacteristic
255#127C4×D6111yescharacteristic
256#126—1224no—
257#126—1224no—
258#125—612no—
259#125—1224no—
260#124—1224no—
261#124C4×D6112yes—
262#123—1224no—
263#123—1224no—
264#123—1222no—
265C22⋊Q8—614no—
266C22⋊Q8—1224no—
267C22⋊Q8—1222no—
268#121—22no—
269#121—614no—
270#121—1224no—
271#120—1222no—
272C4×Q8—614no—
273C4×Q8—1222no—
274C4×D4—22no—
275C4×D4—614no—
276C4×D4—1224no—
277C4×D4—1224no—
278#117—612no—
279#117—612no—
280#117—614no—
281#117—612no—
282#117—1222no—
283C2×C4⋊C4—612no—
284C2×C4⋊C4—1222no—
285C2×C22⋊C4—612no—
286C2×C22⋊C4—1222no—
287C2×C42—22no—
288C2×C42—611no—
289C2×C42—612no—
290C2×C42—1222no—
291C2×C42C22×D6112yes—
292C2×C42C22×D6111yescharacteristic
293C4⋊C8—22no—
294C4⋊C8—1224no—
295C4⋊C8—1222no—
296#104—44no—
297#104—2444no—
298#95—1222no—
299#95—1222no—
300#95—1222no—
301C24—611no—
302#41—24no—
303#41—24no—
304C2×Q8—1222no—
305C2×Q8C2×C4×D6112yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—1224no—
309C2×D4C2×C4×D6112yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—611no—
313C22×C4—612no—
314C22×C4—612no—
315C22×C4—612no—
316C22×C4—1224no—
317C22×C4—1222no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4D4×D6112yes—
321C22×C4C23×D6111yescharacteristic
322M16—24no—
323M16—1224no—
324M16—2444no—
325C2×C8—22no—
326C2×C8—1222no—
327C2×C8—2444no—
328C4⋊C4—22no—
329C4⋊C4—614no—
330C4⋊C4—614no—
331C4⋊C4—614no—
332C4⋊C4—1224no—
333C4⋊C4—1224no—
334C4⋊C4—1224no—
335C4⋊C4—1222no—
336C4⋊C4—2444no—
337C22⋊C4—24no—
338C22⋊C4—614no—
339C22⋊C4—614no—
340C22⋊C4—1224no—
341C22⋊C4—1224no—
342C22⋊C4—1224no—
343C22⋊C4—1222no—
344C22⋊C4—2444no—
345C42—22no—
346C42—22no—
347C42—44no—
348C42—612no—
349C42—612no—
350C42—1224no—
351C42—1222no—
352C42—1222no—
353C42—1222no—
354C42C2×C4×D6111yescharacteristic
355C42C2×C4×D6111yescharacteristic
356C42D4×D6112yes—
357C23—22no—
358C23—612no—
359C23—1222no—
360C23—2442no—
361C23C2×D4×D6111yescharacteristic
362Q8—24no—
363Q8—4882no—
364D4—24no—
365D4—44no—
366D4—82no—
367D4—4884no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—614no—
376C2×C4—612no—
377C2×C4—1222no—
378C2×C4—1224no—
379C2×C4—1222no—
380C2×C4—1222no—
381C2×C4—1224no—
382C2×C4—4884no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4C22⋊C4×D6112yes—
388C2×C4C22⋊C4×D6112yes—
389C2×C4C22×C4×D6111yescharacteristic
390C2×C4C2×D4×D6111yescharacteristic
391C2×C4#1451982212yes—
392C2×C4#1451982412yes—
393C8—42no—
394C8—2442no—
395C22—22no—
396C22—44no—
397C22—42no—
398C22—1224no—
399C22—1222no—
400C22—4882no—
401C22C2×C22⋊C4×D6111yescharacteristic
402C22#1684034112yes—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—42no—
407C4—1224no—
408C4—1222no—
409C4—4882no—
410C4—2444no—
411C4#1684024112yes—
412C4#1684032511yescharacteristic
413C4#1684033211yescharacteristic
414C2—22no—
415C2—82no—
416C2—2442no—
417C2#2144840911yescharacteristic
418C2#2145436312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1962C2
socleC22×C61C22×C22⋊C4
prime radical#21459208C2
cosocleC22×C4×C61C24
hypercenter#1962D61
hypocenterC61#21858
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #21454381C2semidirect
3. #21448316C2semidirect
4. #21446799C2semidirect
5. #21446796C2semidirect
6. #21446652C2semidirect
7. #21446118C2semidirect
8. #21445837C2semidirect
9. C61×#1962C2semidirect
11. #16840381C4semidirect
12. #16840373C4semidirect
13. #16840366C4semidirect
211. #1962D61semidirect
245. C61#21858semidirect

Maximal subgroups

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

subgroupindex
#214592082
#214543812
#214483162
#214467992
#214467962
#214466522
#214461182
#214458372
C61×#19622
#2185861