Finite Group Databasebeta

← #30620424

Subgroups — #30620424

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462620C212yesmaximal, complemented
2#21462012C214yesmaximal, complemented
3#21462011C212yesmaximal, complemented
4#21461684C212yesmaximal
5#21461683C211yesmaximal, characteristic
6#21461569C212yesmaximal
7C4×#14519837C211yesmaximal, characteristic
8C2×#16840425C211yesmaximal, characteristic
9#16840814—24no—
10#16840812—22no—
11#16840811—22no—
12#16840765C2212yes—
13#16840765C2214yes—
14C2×#14519837C2211yescharacteristic
15C2×#14519837C2212yes—
16C2×#14519837C2211yescharacteristic
17C4×#11758217C2212yes—
18C4×#11758217C2214yes—
19#16840642C2212yes—
20#16840642C2214yes—
21#16840642C2212yes—
22#16840642C2214yes—
23C23×Dic122C411yescharacteristic
24C22×Q8×D61C411yescharacteristic
25#16840425—22no—
26#16840425—22no—
27#16840425—24no—
28C2×#14519776C2212yes—
29C2×#14519775C2211yescharacteristic
30C2×#14519757C411yescharacteristic
31C2×#14519755C412yes—
32C2×#14519754C412yes—
33C2×#14519742C2211yescharacteristic
34C2×C4×C22⋊D61C2211yescharacteristic
35C2×#14519737C2212yes—
36C2×C22⋊Q8×C61C411yescharacteristic
37C2×#11758217C2312yes—
38C2×#11758217C2314yes—
39C2×#11758217C2312yes—
40#14519844—24no—
41#14519844—22no—
42C4⋊#92420—22no—
43C4⋊#92420—24no—
44#14519837—22no—
45#14519837—24no—
46#14519837Q814yescomplemented
47#14519832—24no—
48#14519832—22no—
49C4×#92420—22no—
50C4×#92420—24no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—44no—
55C22×Dic122C2×C411yescharacteristic
56C22×Dic122C2×C412yes—
57C22×Dic122C2×C412yes—
58C22×Dic122C2×C411yescharacteristic
59C2×Q8×D61—21no—
60C2×Q8×D61—22no—
61C2×Q8×D61—21no—
62C2×Q8×D61—44no—
63C22×C22⋊D61C2311yescharacteristic
64C23×Dic61C2×C412yes—
65C22×C4×D61C2×C412yes—
66C22×C4×D61C2×C411yescharacteristic
67#14519776—24no—
68#14519776—24no—
69#14519775—22no—
70#14519775—22no—
71#14519757—22no—
72#14519757—22no—
73#14519755—24no—
74#14519755—24no—
75#14519754—24no—
76#14519754—24no—
77#14519742D412yes—
78#14519742D412yes—
79#14519742D414yes—
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82#14519737—24no—
83#14519737—24no—
84C2×C4⋊Dic61C2×C412yes—
85C2×#11758196C2×C412yes—
86C2×#11758196C2×C412yes—
87C2×#11758196C2312yes—
88C2×#11758196C2311yescharacteristic
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C411yescharacteristic
92C2×C4×Dic61C2312yes—
93C2×C4×Dic61C2311yescharacteristic
94C22×Q8×C61C2×C411yescharacteristic
95C23×C4×C61C2×C411yescharacteristic
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C2×C4⋊C4×C61C2×C412yes—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C22⋊C4×C61C2×C412yes—
101C24×C61C22×C411yescharacteristic
102C23×D61C22×C411yescharacteristic
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—42no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—42no—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C412yes—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61C2×Q812yes—
116C2×C22⋊D61#4112yes—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—44no—
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C412yes—
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C2411yescharacteristic
125Q8×D61—44no—
126Q8×D61—48no—
127Q8×D61—44no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—44no—
133C2×C4×D61—42no—
134C2×Dic122—24no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—44no—
142C2×Dic122—48no—
143C2×Dic122—44no—
144C2×Dic122—44no—
145C2×Dic122—44no—
146C2×Dic122—42no—
147C2×Dic122C22⋊C411yescharacteristic
148C2×Dic122C22⋊C412yes—
149C2×Dic122C22⋊C411yescharacteristic
150#11758217—22no—
151#11758217—22no—
152#11758217—24no—
153#11758217—22no—
154#11758217—24no—
155#11758217C2×Q814yescomplemented
156C2×#92420—22no—
157C2×#92420—22no—
158C2×#92420—24no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61C22⋊C412yes—
161C4⋊C4×C61C22⋊C412yes—
162C22⋊C4×C61—24no—
163C22⋊Dic61—22no—
164#11758198—24no—
165#11758198—22no—
166C4⋊Dic61—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—22no—
170#11758196C22⋊C414yes—
171#11758196C22⋊C414yes—
172C4×Dic61C2×D412yes—
173C4×Dic61C2×D414yes—
174C4×Dic61C2×D414yes—
175C4×Dic61C2×D412yes—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C4×Q812yes—
181C23×C61C23×C411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61C4×Q812yes—
187Q8×C61—42no—
188Q8×C61—42no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61C2×C22⋊C414yes—
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C22⋊D61—24no—
201C22⋊D61—24no—
202C2×Dic61—24no—
203C2×Dic61—22no—
204C2×Dic61—21no—
205C2×Dic61—41no—
206C2×Dic61—44no—
207C2×Dic61—44no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61C2×C22⋊C412yes—
211C2×Dic61C2×C22⋊C412yes—
212C2×Dic61C2×C22⋊C412yes—
213C2×Dic61C22×D412yes—
214C2×Dic61C22×Q811yescharacteristic
215C2×Dic61C2×#4111yescharacteristic
216C2×Dic612−1+412yes—
217C4×D61—44no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—44no—
221Dic122—22no—
222Dic122—24no—
223Dic122—44no—
224Dic122—44no—
225Dic122—42no—
226Dic122—42no—
227Dic122—44no—
228Dic122—44no—
229Dic122#10014yescomplemented
230#92420—24no—
231#92420—42no—
232#92420—44no—
233Q8×#100—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61C22×C22⋊C411yescharacteristic
242C22×C61C2×C4×Q811yescharacteristic
243C22×C61#51911yescharacteristic
244C2×D61—22no—
245C2×D61—42no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61—42no—
249C4×C61—21no—
250C4×C61—24no—
251C4×C61—41no—
252C4×C61—42no—
253C4×C61C2×#10012yes—
254Dic61—22no—
255Dic61—22no—
256Dic61—41no—
257Dic61—42no—
258Dic61C2×#10014yes—
259Dic61#54812yes—
260Dic61D4×Q812yes—
261C2×D4×Q8—611no—
262Q8×M16—612no—
263#1997—612no—
264#1997—614no—
265#1800—611no—
266#1800—612no—
267C4×#100—611no—
268C4×#100—612no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—42no—
272C2×C61Q8×C22⋊C411yescharacteristic
273C2×C61C22×#10011yescharacteristic
274C2×C61#288911yescharacteristic
275D61—42no—
276C23×Q8—611no—
277C23×Q8—611no—
278D4×Q8—1222no—
279D4×Q8—1224no—
280D4×Q8—1222no—
281C2×C4⋊Q8—611no—
282C2×C4⋊Q8—612no—
283C2×C22⋊Q8—612no—
284C2×C22⋊Q8—611no—
285C2×C22⋊Q8—612no—
286C2×C22⋊Q8#692311yescharacteristic
287C2×C4×Q8—611no—
288C2×C4×D4—611no—
289C2×C4×D4—612no—
290#446—1222no—
291#446—1224no—
292C8×Q8—1222no—
293#423—612no—
294#423—614no—
295C2×#100—611no—
296C2×#100—612no—
297C2×#100—611no—
298C4×M16—612no—
299C4×M16—614no—
300#343—612no—
301#343—614no—
302#343—612no—
303#343—614no—
304C61Q8×#10011yescharacteristic, complemented, Sylow
305C22×Q8—611no—
306C22×Q8—612no—
307C22×Q8—612no—
308C22×Q8—611no—
309C22×Q8—1221no—
310C22×Q8—1222no—
311C22×Q8—1221no—
312C22×Q8—1221no—
313C22×Q8—1222no—
314C22×Q8—1221no—
315C22×Q8—2444no—
316C22×Q8—2444no—
317C22×Q8C2×#692311yescharacteristic
318C22×D4—611no—
319C23×C4—612no—
320C23×C4—612no—
321C23×C4—611no—
322C23×C4C2×#692311yescharacteristic
323C2×M16—612no—
324C2×M16—614no—
325C2×M16—612no—
326C4⋊Q8—1224no—
327C4⋊Q8—1224no—
328C4⋊Q8—1222no—
329C4⋊Q8—1222no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—1224no—
333C22⋊Q8—1224no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336C22⋊Q8—1222no—
337C22⋊Q8—1222no—
338C4×Q8—612no—
339C4×Q8—612no—
340C4×Q8—614no—
341C4×D4—1224no—
342C4×D4—1224no—
343C4×D4—1222no—
344C4×D4—1222no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4C2×#692312yes—
351C2×C4⋊C4C2×#692311yescharacteristic
352C2×C22⋊C4—611no—
353C2×C22⋊C4—611no—
354C2×C22⋊C4—612no—
355C2×C22⋊C4C2×#692312yes—
356C2×C42—612no—
357C2×C42—611no—
358C4⋊C8—1222no—
359C4⋊C8—1224no—
360C4⋊C8—1224no—
361C4⋊C8—1222no—
362#100—614no—
363#100—1222no—
364#100—1224no—
365#97—1224no—
366#97—1222no—
367C4×C8—1222no—
368C4×C8—1224no—
369C24—611no—
370C24C22×#692311yescharacteristic
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—42no—
374C2×Q8—611no—
375C2×Q8—612no—
376C2×Q8—611no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C2×Q8—1224no—
382C2×Q8—1221no—
383C2×Q8—1221no—
384C2×Q8—2442no—
385C2×Q8—2444no—
386C2×Q8—2444no—
387C2×Q8—2444no—
388C2×Q8—2448no—
389C2×Q8—2444no—
390C2×Q8—2444no—
391C2×Q8—2448no—
392C2×Q8—2444no—
393C2×D4—612no—
394C2×D4—612no—
395C2×D4—1222no—
396C2×D4—1222no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—611no—
401C22×C4—612no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—612no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—2442no—
412C22×C4—2444no—
413C22×C4—2444no—
414C22×C4C22×#692311yescharacteristic
415C22×C4C22×#692311yescharacteristic
416C22×C4C22×#692312yes—
417C22×C4C22×#692312yes—
418M16—614no—
419M16—1222no—
420M16—1224no—
421M16—1222no—
422M16—1224no—
423M16—1222no—
424C2×C8—1222no—
425C2×C8—1224no—
426C2×C8—1222no—
427C4⋊C4—24no—
428C4⋊C4—614no—
429C4⋊C4—614no—
430C4⋊C4—1224no—
431C4⋊C4—1222no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4C22⋊#692312yes—
435C4⋊C4C22⋊#692312yes—
436C22⋊C4—24no—
437C22⋊C4—1222no—
438C22⋊C4—1222no—
439C22⋊C4—1224no—
440C42—614no—
441C42—612no—
442C42—612no—
443C42—614no—
444C23—21no—
445C23—21no—
446C23—41no—
447C23—41no—
448C23—612no—
449C23—1221no—
450C23—1221no—
451C23—2441no—
452C23—2441no—
453C23Q8×#692312yes—
454C23C23×#692311yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—614no—
458Q8—1224no—
459Q8—1222no—
460Q8—2442no—
461Q8—2444no—
462Q8—2444no—
463Q8—2444no—
464Q8—2444no—
465Q8—2442no—
466D4—1224no—
467D4—1224no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—611no—
479C2×C4—611no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—1224no—
484C2×C4—1222no—
485C2×C4—1221no—
486C2×C4—2442no—
487C2×C4—2444no—
488C2×C4—2444no—
489C2×C4—2444no—
490C2×C4—2441no—
491C2×C4—2444no—
492C2×C4—2442no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4C2×C22⋊#692314yes—
496C2×C4C2×C22⋊#692311yescharacteristic
497C2×C4C2×C22⋊#692311yescharacteristic
498C8—1224no—
499C8—2444no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—1222no—
509C22—2442no—
510C22—2442no—
511C22—2441no—
512C22—2441no—
513C22C22×C22⋊#692311yescharacteristic
514C22C2×Q8×#692311yescharacteristic
515C22#1684083011yescharacteristic
516C4—21no—
517C4—24no—
518C4—41no—
519C4—42no—
520C4—612no—
521C4—612no—
522C4—614no—
523C4—1222no—
524C4—1222no—
525C4—2441no—
526C4—2442no—
527C4D61⋊#10012yes—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—2442no—
532C2Q8×C22⋊#692311yescharacteristic
533C2C2×D61⋊#10011yescharacteristic
534C2#2146243111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61Q8×#100
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462620C2semidirect
2. #21462012C2semidirect
3. #21462011C2semidirect
46. #14519837Q8semidirect
155. #11758217C2×Q8semidirect
229. Dic122#100semidirect
304. C61Q8×#100semidirect

Maximal subgroups

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

subgroupindex
#214626202
#214620122
#214620112
#214616842
#214616832
#214615692
C4×#145198372
C2×#168404252
Q8×#10061