Finite Group Databasebeta

← #30610454

Subgroups — #30610454

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic
2#21462495C211yesmaximal, characteristic
3#21461161C214yesmaximal, complemented
4#21461147C214yesmaximal, complemented
5#21461121C214yesmaximal, complemented
6C22×Q8×Dic61C211yesmaximal, characteristic
7#16840783—21no—
8#16840783C2212yes—
9#16840782—21no—
10#16840782—21no—
11#16840779—21no—
12#16840779C2212yes—
13#16840778C2212yes—
14#16840778C2212yes—
15#16840765—21no—
16#16840765C2212yes—
17#16840732—21no—
18#16840732—21no—
19C4×#11758217—21no—
20C4×#11758217C2212yes—
21Q8⋊#92420C22116yescomplemented
22C2×Q8×Dic61—21no—
23C2×Q8×Dic61—21no—
24C2×Q8×Dic61—21no—
25C2×Q8×Dic61—21no—
26C2×Q8×Dic61—21no—
27C2×Q8×Dic61—21no—
28C2×Q8×Dic61C412yes—
29C2×Q8×Dic61C412yes—
30C2×Q8×Dic61C412yes—
31C2×Q8×Dic61C2212yes—
32C2×Q8×Dic61C2212yes—
33C2×Q8×Dic61C2212yes—
34C22×C4⋊Dic61—21no—
35C22×C4⋊Dic61C411yescharacteristic
36C22×C4×Dic61—21no—
37C22×C4×Dic61C2211yescharacteristic
38C23×Q8×C61C411yescharacteristic
39C2×#11758217—22no—
40C2×#11758217—22no—
41#14519844—22no—
42#14519844—22no—
43C4⋊#92420—22no—
44C4⋊#92420C2314yes—
45C22⋊#92420—22no—
46C22⋊#92420—22no—
47C22⋊#92420—22no—
48C22⋊#92420—22no—
49#14519832—22no—
50#14519832—22no—
51C4×#92420—22no—
52C4×#92420C2314yes—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56Q8×Dic61—22no—
57Q8×Dic61—22no—
58Q8×Dic61—22no—
59Q8×Dic61—22no—
60Q8×Dic61—22no—
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61C2×C414yes—
69Q8×Dic61C2×C414yes—
70Q8×Dic61C2×C414yes—
71Q8×Dic61C2314yes—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61C2×C412yes—
82C2×C4⋊Dic61C2×C412yes—
83C2×C4⋊Dic61C2×C412yes—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61—22no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61D411yescharacteristic
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C22×Q8×C61—28no—
103C22×Q8×C61—21no—
104C22×Q8×C61—21no—
105C22×Q8×C61—21no—
106C22×Q8×C61C2×C412yes—
107C22×Q8×C61C2×C412yes—
108C22×Q8×C61C2×C412yes—
109C23×C4×C61—21no—
110C23×C4×C61C2×C411yescharacteristic
111C24×C61C22⋊C411yescharacteristic
112C2×Q8×C61—28no—
113C2×Q8×C61—28no—
114C2×Q8×C61—28no—
115C2×Q8×C61—28no—
116C2×Q8×C61—28no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—24no—
121C2×Q8×C61M1618yescomplemented
122C2×Q8×C61C22×C414yes—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—24no—
128C22×C4×C61C22⋊C411yescharacteristic
129C22×C4×C61C22⋊C411yescharacteristic
130C22×C4×C61C22⋊C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—42no—
139C22×Dic61—44no—
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758217—44no—
153#11758217—44no—
154C2×#92420—24no—
155C2×#92420—24no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61C22×C414yes—
164C4×Dic61—24no—
165C4×Dic61—24no—
166C4×Dic61—24no—
167C4×Dic61—24no—
168C4×Dic61C22⋊C411yescharacteristic
169C4×Dic61C22⋊C411yescharacteristic
170C4×Dic61C22⋊C411yescharacteristic
171C4×Dic61C22⋊C411yescharacteristic
172C4×Dic61C22⋊C411yescharacteristic
173C4×Dic61C22⋊C411yescharacteristic
174C4×Dic61C22×C411yescharacteristic
175C4×Dic61C22×C411yescharacteristic
176C4×Dic61C22×C411yescharacteristic
177C4×Dic61C2×D411yescharacteristic
178C4×Dic61C2×D411yescharacteristic
179C4×Dic61C2×D411yescharacteristic
180C4×Dic61C2×D411yescharacteristic
181C4×Dic61C2×D411yescharacteristic
182C4×Dic61C2×D411yescharacteristic
183C4×Dic61C2411yescharacteristic
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61C2×C22⋊C411yescharacteristic
188C23×C61C2×C22⋊C411yescharacteristic
189C23×C61C2×C22⋊C411yescharacteristic
190Q8×C61—28no—
191Q8×C61—28no—
192Q8×C61—24no—
193Q8×C61—28no—
194Q8×C61C2×M1618yes—
195C2×C4×C61—28no—
196C2×C4×C61—28no—
197C2×C4×C61—24no—
198C2×C4×C61—28no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61C2×M1614yes—
208C2×C4×C61C23×C411yescharacteristic
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—24no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—44no—
222C2×Dic61—44no—
223C2×Dic61—44no—
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C2×C22⋊C411yescharacteristic
227C2×Dic61C2×C22⋊C411yescharacteristic
228C2×Dic61C2×C22⋊C411yescharacteristic
229C2×Dic61C2×C22⋊C411yescharacteristic
230C2×Dic61#13714yes—
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C22×D411yescharacteristic
233#92420—44no—
234#92420—44no—
235#17449—611nomaximal, Sylow
236C22×C61—28no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61#40712yes—
244C22×C61#41912yes—
245C22×C61#42112yes—
246C22×C61C22×C22⋊C411yescharacteristic
247C4×C61—28no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61#40712yes—
251C4×C61C22×M1612yes—
252Dic61—21no—
253Dic61—21no—
254Dic61—21no—
255Dic61—21no—
256Dic61—21no—
257Dic61—21no—
258Dic61—44no—
259Dic61#41912yes—
260Dic61C2×#13712yes—
261C22×C4×Q8—611no—
262#2920—611no—
263#2920—611no—
264#1491—614no—
265#1487—614no—
266#1479—614no—
267C2×C61—22no—
268C2×C61—22no—
269C2×C61—22no—
270C2×C61C2×#40711yescharacteristic
271C2×C61C2×#41911yescharacteristic
272C2×C61C2×#42111yescharacteristic
273C23×Q8#692311yescharacteristic
274C2×C4×Q8—612no—
275C2×C4×Q8—612no—
276C2×C4×Q8—612no—
277C2×C4×Q8—612no—
278C2×C4×Q8—612no—
279C2×C4×Q8—612no—
280C2×C4×Q8—1221no—
281C2×C4×Q8—1221no—
282C2×C4×Q8—1221no—
283C2×C4×Q8—1221no—
284C2×C4×Q8—1221no—
285C2×C4×Q8—1221no—
286C22×C4⋊C4—611no—
287C22×C4⋊C4—1221no—
288C22×C42—611no—
289C22×C42—1221no—
290#432—612no—
291#432—1221no—
292#432—1221no—
293#432—1221no—
294#431—612no—
295#431—612no—
296#431—612no—
297#431—1221no—
298#423—612no—
299#423—1221no—
300#407—1221no—
301#407—1221no—
302C4×M16—612no—
303C4×M16—1221no—
304Q8⋊C8—6116no—
305C61#1744911yescharacteristic, complemented, Sylow
306C22×Q8—28no—
307C22×Q8—21no—
308C22×Q8—21no—
309C22×Q8—21no—
310C22×Q8C2×#692312yes—
311C22×Q8C2×#692312yes—
312C22×Q8C2×#692312yes—
313C23×C4—21no—
314C23×C4—611no—
315C23×C4—611no—
316C23×C4—1221no—
317C23×C4C2×#692311yescharacteristic
318C2×M16—1222no—
319C2×M16—1222no—
320C4×Q8—614no—
321C4×Q8—614no—
322C4×Q8—614no—
323C4×Q8—614no—
324C4×Q8—1222no—
325C4×Q8—1222no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×Q8—1222no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—1222no—
361C2×C42—1222no—
362C2×C42—1222no—
363C2×C42—1222no—
364C2×C42—1222no—
365C2×C42—1222no—
366C4⋊C8—614no—
367C4⋊C8—1222no—
368C4⋊C8—1222no—
369C4⋊C8—1222no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97—1222no—
375#97—1222no—
376C4×C8—614no—
377C4×C8—1222no—
378C24C22⋊#692311yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8#1175821718yescomplemented
389C2×Q8C22×#692314yes—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—1224no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—2442no—
412C22×C4—2444no—
413C22×C4C22⋊#692311yescharacteristic
414C22×C4C22⋊#692311yescharacteristic
415C22×C4C22⋊#692311yescharacteristic
416C22×C4C22×#692311yescharacteristic
417C22×C4C22×#692311yescharacteristic
418C22×C4C22×#692311yescharacteristic
419M16—2444no—
420M16—2444no—
421C2×C8—1224no—
422C2×C8—1224no—
423C4⋊C4—614no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C42—611no—
432C42—611no—
433C42—611no—
434C42—611no—
435C42—611no—
436C42—611no—
437C42—611no—
438C42—611no—
439C42—611no—
440C42—611no—
441C42—611no—
442C42—611no—
443C42—611no—
444C42—611no—
445C42—611no—
446C42—611no—
447C42—1224no—
448C42—1224no—
449C42—1224no—
450C42—1224no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23C2×C22⋊#692311yescharacteristic
455C23C2×C22⋊#692311yescharacteristic
456C23C2×C22⋊#692311yescharacteristic
457Q8—28no—
458Q8—28no—
459Q8—28no—
460Q8—24no—
461Q8C2×#1175821718yes—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—614no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—611no—
478C2×C4—611no—
479C2×C4—611no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1224no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2444no—
493C2×C4—2444no—
494C2×C4—2444no—
495C2×C4C2×C22⋊#692311yescharacteristic
496C2×C4C2×C22⋊#692311yescharacteristic
497C2×C4C2×C22⋊#692311yescharacteristic
498C2×C4C2×#1175821714yes—
499C2×C4C23×#692311yescharacteristic
500C8—2444no—
501C8—2444no—
502C22—28no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22#1684073212yes—
510C22#1684075512yes—
511C22#1684076112yes—
512C22C22×C22⋊#692311yescharacteristic
513C4—28no—
514C4—21no—
515C4—21no—
516C4—612no—
517C4—612no—
518C4—1221no—
519C4—1221no—
520C4—1221no—
521C4—1221no—
522C4—1221no—
523C4—1221no—
524C4—2444no—
525C4#1684073212yes—
526C4C22×#1175821712yes—
527C2—22no—
528C2—22no—
529C2—22no—
530C2C2×#1684073211yescharacteristic
531C2C2×#1684075511yescharacteristic
532C2C2×#1684076111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC23×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC4×Dic61C24
hypercenterC23×Q8#6923
hypocenterC61#17449
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21461161C2semidirect
4. #21461147C2semidirect
5. #21461121C2semidirect
21. Q8⋊#92420C22semidirect
121. C2×Q8×C61M16semidirect
305. C61#17449semidirect
388. C2×Q8#11758217semidirect

Maximal subgroups

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

subgroupindex
#214624952
#214611612
#214611472
#214611212
C22×Q8×Dic612
#1744961