Finite Group Databasebeta

← #30598200

Subgroups — #30598200

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461121C212yesmaximal, complemented
2#21461120C212yesmaximal, complemented
3#21461052C211yesmaximal, characteristic, complemented
4#21461052C211yesmaximal, characteristic, complemented
5C2×#16840244C211yesmaximal, characteristic
6#16840778C2212yes—
7#16840778C2212yes—
8Q8⋊#92420—24no—
9#16840610—24no—
10#16840608—21no—
11#16840608—21no—
12#16840606—21no—
13#16840606—21no—
14C22×#11758226C411yescharacteristic
15#16840244—24no—
16#16840244—24no—
17C2×Q8×Dic61C2211yescharacteristic
18C2×C4×D244—21no—
19C2×C4×D244C411yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×#14519734—21no—
23C2×#14519734C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C411yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C4×#92420—22no—
32C4×#92420—22no—
33C2×#11758226—22no—
34C2×#11758226—22no—
35C2×#11758226—22no—
36C2×#11758226—22no—
37C2×#11758226—22no—
38C2×#11758226C814yescomplemented
39C22×D244—21no—
40C22×D244C2×C411yescharacteristic
41C22×C4×D61—21no—
42C22×C4×D61—21no—
43C22×C4×D61—21no—
44C22×C4×D61—21no—
45C22×C4×D61C2×C411yescharacteristic
46Q8×Dic61—22no—
47Q8×Dic61D414yes—
48C4×D244—28no—
49C4×D244—22no—
50C4×D244—22no—
51#14519739—28no—
52#14519739—22no—
53#14519739D414yes—
54#14519734—28no—
55#14519734—22no—
56#14519734—22no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2×C411yescharacteristic
59C2×#11758196—21no—
60C2×#11758196D411yescharacteristic
61C2×#11758196D411yescharacteristic
62C4⋊C4×D61—28no—
63C4⋊C4×D61—22no—
64C4⋊C4×D61—22no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198D411yescharacteristic
68C2×#11758198D411yescharacteristic
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2311yescharacteristic
71C22×Q8×C61C2×C411yescharacteristic
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C42×C61—21no—
77C2×C42×C61C2×C411yescharacteristic
78C23×D61—21no—
79C23×D61C22⋊C411yescharacteristic
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61C22⋊C412yes—
84C2×Q8×C61C2×C812yes—
85C2×Q8×C61M1612yes—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22×C411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C2×D411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95#11758226—28no—
96#11758226—28no—
97#11758226—28no—
98#11758226—28no—
99C2×D244—24no—
100C2×D244—24no—
101C2×D244—24no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C22⋊C412yes—
106C2×D244C2×C812yes—
107C2×D244M1612yes—
108C2×C4×D61—24no—
109C2×C4×D61—24no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—24no—
117C2×C4×D61—24no—
118C2×C4×D61—82no—
119C2×C4×D61—82no—
120C2×C4×D61—82no—
121C2×C4×D61C2×C812yes—
122C2×C4×D61M1612yes—
123C2×#92420—42no—
124C2×#92420—42no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61C22⋊C412yes—
128C42×C61—24no—
129C42×C61—21no—
130C42×C61—21no—
131#11758198—24no—
132#11758198—24no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—24no—
137C4⋊Dic61C22⋊C412yes—
138#11758196—24no—
139#11758196—22no—
140#11758196—22no—
141C4×Dic61—24no—
142C4×Dic61C2×D412yes—
143C4×Dic61C2×D412yes—
144C23×C61C2×C22⋊C411yescharacteristic
145C22×D61—21no—
146C22×D61—22no—
147C22×D61—22no—
148C22×D61—21no—
149C22×D61—22no—
150C22×D61—42no—
151C22×D61—42no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61C22⋊C812yes—
155Q8×C61—22no—
156Q8×C61—22no—
157Q8×C61—22no—
158Q8×C61C22⋊C814yes—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C22×C811yescharacteristic
171C2×C4×C61C2×M1611yescharacteristic
172C2×C4×C61C2×M1611yescharacteristic
173C2×C4×C61C2×M1611yescharacteristic
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—22no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—42no—
180C2×Dic61C22⋊C812yes—
181C2×Dic61#12012yes—
182C2×Dic61#12012yes—
183D244—216no—
184D244—22no—
185D244—22no—
186D244—22no—
187D244C22⋊C814yes—
188C4×D61—216no—
189C4×D61—22no—
190C4×D61—22no—
191C4×D61—22no—
192C4×D61—22no—
193C4×D61—84no—
194C4×D61—84no—
195C4×D61—84no—
196#92420—82no—
197#92420—82no—
198#8324—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—21no—
201C22×C61#37911yescharacteristic
202C22×C61C2×C22⋊C811yescharacteristic
203C22×C61#40711yescharacteristic
204C2×D61—22no—
205C2×D61—24no—
206C2×D61—42no—
207C2×D61—44no—
208C2×D61—44no—
209C2×D61—41no—
210C2×D61—42no—
211C2×D61—41no—
212C2×D61—42no—
213C2×D61—42no—
214C4×C61—24no—
215C4×C61—41no—
216C4×C61—42no—
217C4×C61—41no—
218C4×C61C2×C22⋊C811yescharacteristic
219C4×C61C2×C22⋊C811yescharacteristic
220C4×C61#40711yescharacteristic
221C4×C61#40711yescharacteristic
222Dic61—22no—
223Dic61—22no—
224Dic61—22no—
225Dic61—42no—
226C2×#520—611no—
227#1479—612no—
228#1478—612no—
229#1458—611no—
230#1458—611no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61#177911yescharacteristic
234C2×C61#179511yescharacteristic
235C2×C61#179611yescharacteristic
236D61—42no—
237D61—44no—
238C22×#41—611no—
239#520—1224no—
240#520—1224no—
241C2×C4×Q8—611no—
242C2×C4×Q8#692311yescharacteristic
243C2×C4×D4—611no—
244C2×C4×D4—611no—
245C2×C4×D4—1221no—
246C2×C4×D4—1221no—
247C2×#117—611no—
248C2×#117—1221no—
249C22×C4⋊C4—611no—
250C22×C4⋊C4—1221no—
251#431—612no—
252#431—612no—
253Q8⋊C8—1224no—
254D4⋊C8—1224no—
255#324—1221no—
256#324—1221no—
257#323—1221no—
258#323—1221no—
259C61#832411yescharacteristic, complemented, Sylow
260C2×#41—614no—
261C2×#41—1222no—
262C2×#41—1222no—
263C2×#41—1222no—
264C2×#41—1222no—
265C2×#41—1222no—
266C22×Q8C2×#692311yescharacteristic
267C22×D4—611no—
268C22×D4—1221no—
269C23×C4—611no—
270C23×C4—1221no—
271C23×C4—1221no—
272C23×C4—1221no—
273C23×C4—1221no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—614no—
277C4×Q8—1222no—
278C4×D4—614no—
279C4×D4—1222no—
280C4×D4—1222no—
281C4×D4—1228no—
282C4×D4—1222no—
283C4×D4—1228no—
284#117—1222no—
285#117—1222no—
286#117—1228no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—611no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—611no—
291C2×C4⋊C4—1222no—
292C2×C4⋊C4—1228no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1222no—
296C2×C4⋊C4C2×#692311yescharacteristic
297C2×C22⋊C4—611no—
298C2×C22⋊C4—611no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C42—21no—
302C2×C42—611no—
303C2×C42—1221no—
304C2×C42C2×#692311yescharacteristic
305C4⋊C8—1222no—
306C4⋊C8—1222no—
307C22⋊C8—1222no—
308C22⋊C8—1222no—
309C4×C8—1222no—
310C4×C8—1222no—
311C24—611no—
312C24—1221no—
313#41—1228no—
314#41—1228no—
315#41—1228no—
316#41—1228no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8D61⋊C812yes—
321C2×Q8#1175821312yes—
322C2×Q8C22⋊#692312yes—
323C2×D4—612no—
324C2×D4—612no—
325C2×D4—612no—
326C2×D4—1224no—
327C2×D4—1224no—
328C2×D4—1224no—
329C2×D4—1221no—
330C2×D4—1221no—
331C2×D4—1221no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—611no—
337C22×C4—612no—
338C22×C4—612no—
339C22×C4—1224no—
340C22×C4—1224no—
341C22×C4—1224no—
342C22×C4—1224no—
343C22×C4—1224no—
344C22×C4—1224no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—4882no—
351C22×C4—4882no—
352C22×C4—4882no—
353C22×C4C22⋊#692311yescharacteristic
354C22×C4C22⋊#692311yescharacteristic
355C22×C4C22×#692311yescharacteristic
356C2×C8—2442no—
357C2×C8—2442no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—612no—
361C4⋊C4—1222no—
362C4⋊C4—1222no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1221no—
366C4⋊C4C22⋊#692312yes—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C22⋊C4—1224no—
370C22⋊C4—1224no—
371C42—24no—
372C42—21no—
373C42—21no—
374C42—612no—
375C42—612no—
376C42—1224no—
377C23—612no—
378C23—1222no—
379C23—1222no—
380C23—1221no—
381C23—1222no—
382C23—1221no—
383C23—2441no—
384C23—2441no—
385C23—2442no—
386C23—2442no—
387C23C2×C22⋊#692311yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8#1451983414yes—
392D4—614no—
393D4—12216no—
394D4—1222no—
395D4—1222no—
396D4—1222no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—12216no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—1222no—
417C2×C4—4884no—
418C2×C4—4884no—
419C2×C4—4884no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4C2×C22⋊#692311yescharacteristic
424C2×C4C2×C22⋊#692311yescharacteristic
425C2×C4C2×D61⋊C811yescharacteristic
426C2×C4D61⋊M1611yescharacteristic
427C2×C4D61⋊M1611yescharacteristic
428C2×C4C2×#1175821311yescharacteristic
429C8—4882no—
430C8—4882no—
431C22—21no—
432C22—21no—
433C22—1224no—
434C22—1222no—
435C22—2442no—
436C22—2444no—
437C22—2444no—
438C22—2442no—
439C22—2441no—
440C22—2442no—
441C22—2441no—
442C22—2442no—
443C22#1684070311yescharacteristic
444C22D61⋊(C22⋊C8)11yescharacteristic
445C22#1684073311yescharacteristic
446C4—24no—
447C4—42no—
448C4—41no—
449C4—41no—
450C4—1222no—
451C4—1222no—
452C4—1222no—
453C4—2442no—
454C4D61⋊(C22⋊C8)11yescharacteristic
455C4C2×#1451983411yescharacteristic
456C4#1684073311yescharacteristic
457C4#1684073411yescharacteristic
458C2—21no—
459C2—21no—
460C2—2444no—
461C2—2442no—
462C2#2146164211yescharacteristic
463C2#2146167011yescharacteristic
464C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC22×#11758226C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8324
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21461121C2semidirect
2. #21461120C2semidirect
3. #21461052C2semidirect
4. #21461052C2semidirect
38. C2×#11758226C8semidirect
259. C61#8324semidirect

Maximal subgroups

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

subgroupindex
#214611212
#214611202
#214610522
C2×#168402442
#832461