Finite Group Databasebeta

← #30598159

Subgroups — #30598159

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461674C211yesmaximal, characteristic
2C4×#14519834C211yesmaximal, characteristic
3C2×#16840612C212yesmaximal, complemented
4C2×Q8⋊#92420C212yesmaximal, complemented
5C2×C4×Q8×D61C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×#14519834—21no—
8C2×#14519834—21no—
9C2×C4×#92420C2212yes—
10#16840628—21no—
11#16840628—21no—
12#16840612—24no—
13Q8⋊#92420—24no—
14C22×Q8×D61C411yescharacteristic
15C2×Q8×Dic61C2211yescharacteristic
16C2×C4×Dic122—21no—
17C2×C4×Dic122C411yescharacteristic
18C2×#14519742—21no—
19C2×#14519742C2211yescharacteristic
20C4×Q8×D61—24no—
21C4×Q8×D61—24no—
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C411yescharacteristic
24C2×C42×D61—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic
27C22×#92420—22no—
28C22×#92420—22no—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519834—24no—
32#14519834—42no—
33C4×#92420—22no—
34C4×#92420—22no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C2×Q8×D61—22no—
38C2×Q8×D61—22no—
39C2×Q8×D61—22no—
40C2×Q8×D61—22no—
41C2×Q8×D61—22no—
42C2×Q8×D61C814yescomplemented
43C22×C4×D61—21no—
44C22×C4×D61—21no—
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48Q8×Dic61—22no—
49Q8×Dic61D414yes—
50C4×Dic122—28no—
51C4×Dic122—22no—
52C4×Dic122—22no—
53#14519742—28no—
54#14519742—22no—
55#14519742D414yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C4⋊C4×D61—28no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61—22no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61—28no—
71C42×D61—22no—
72C42×D61—22no—
73C22×Q8×C61C2×C411yescharacteristic
74C4×Q8×C61—22no—
75C4×Q8×C61—22no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61C2×C411yescharacteristic
78C2×C42×C61—21no—
79C2×C42×C61C2×C411yescharacteristic
80C23×D61C22⋊C411yescharacteristic
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61C22⋊C412yes—
85C2×Q8×C61C2×C812yes—
86C2×Q8×C61M1612yes—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22×C411yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97Q8×D61—28no—
98Q8×D61—28no—
99Q8×D61—28no—
100Q8×D61—28no—
101C2×C4×D61—24no—
102C2×C4×D61—24no—
103C2×C4×D61—24no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—24no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—41no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61C2×C812yes—
117C2×C4×D61M1612yes—
118C2×C4×D61SD1612yes—
119C2×C4×D61Q1612yes—
120C2×Dic122—24no—
121C2×Dic122—24no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—24no—
126C2×Dic122C22⋊C412yes—
127C2×Dic122C2×C812yes—
128C2×Dic122M1612yes—
129C2×#92420—24no—
130C2×#92420—42no—
131C2×#92420—42no—
132C2×#92420—42no—
133C2×#92420—42no—
134C2×#92420—42no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C22⋊C412yes—
138C42×C61—24no—
139C42×C61—21no—
140C42×C61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—24no—
143C4⋊Dic61C22⋊C412yes—
144#11758196—24no—
145#11758196—24no—
146#11758196—24no—
147C4×Dic61—24no—
148C4×Dic61—24no—
149C4×Dic61—24no—
150C4×Dic61—22no—
151C4×Dic61—22no—
152C4×Dic61C2×D412yes—
153C4×Dic61C2×D412yes—
154C23×C61C2×C22⋊C411yescharacteristic
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61C22⋊C812yes—
160C22×D61#10312yes—
161C22×D61#10412yes—
162Q8×C61—22no—
163Q8×C61—22no—
164Q8×C61—22no—
165Q8×C61C22⋊C814yescomplemented
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61#10312yes—
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C2×C4×C61C22×C811yescharacteristic
181C2×C4×C61C2×M1611yescharacteristic
182C2×C4×C61C2×M1611yescharacteristic
183C2×C4×C61C2×M1611yescharacteristic
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—21no—
188C2×Dic61—22no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—22no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61C22⋊C812yes—
195C2×Dic61#12012yes—
196C2×Dic61#12012yes—
197C2×Dic61C2×SD1611yescharacteristic
198C2×Dic61C2×Q1611yescharacteristic
199C2×Dic61#13712yes—
200C4×D61—216no—
201C4×D61—22no—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—22no—
205C4×D61—24no—
206C4×D61—24no—
207C4×D61—44no—
208C4×D61—48no—
209Dic122—216no—
210Dic122—22no—
211Dic122—22no—
212Dic122—22no—
213Dic122C22⋊C814yescomplemented
214#92420—44no—
215#92420—82no—
216#8311—611nomaximal, Sylow
217C22×C61#37911yescharacteristic
218C22×C61C2×C22⋊C811yescharacteristic
219C22×C61#40711yescharacteristic
220C22×C61C2×#10311yescharacteristic
221C22×C61#41911yescharacteristic
222C22×C61C2×#10411yescharacteristic
223C22×C61#42111yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61Q8⋊C814yescomplemented
231C4×C61—24no—
232C4×C61—22no—
233C4×C61—41no—
234C4×C61—42no—
235C4×C61C2×C22⋊C811yescharacteristic
236C4×C61C2×C22⋊C811yescharacteristic
237C4×C61#40711yescharacteristic
238C4×C61#40711yescharacteristic
239Dic61—22no—
240Dic61—24no—
241Dic61—22no—
242Dic61—42no—
243Dic61#44812yes—
244Dic61#45112yes—
245C22×C4×Q8—611no—
246#1797—611no—
247C4×C22⋊C8—611no—
248C2×Q8⋊C8—612no—
249C2×Q8⋊C8—612no—
250C2×C61C2×Q8⋊C811yescharacteristic
251C2×C61#147911yescharacteristic
252C2×C61#148711yescharacteristic
253C2×C61#149111yescharacteristic
254C2×C61#177911yescharacteristic
255C2×C61#178711yescharacteristic
256C2×C61#178911yescharacteristic
257D61—24no—
258C23×Q8—611no—
259C2×C4×Q8—611no—
260C2×C4×Q8—611no—
261C2×C4×Q8—611no—
262C2×C4×Q8—1221no—
263C2×C4×Q8—1221no—
264C2×C4×Q8—1224no—
265C2×C4×Q8—1224no—
266C2×C4×Q8#692311yescharacteristic
267C22×C4⋊C4—611no—
268C22×C4⋊C4—1221no—
269C22×C42—611no—
270C22×C42—1221no—
271C2×C4⋊C8—612no—
272C2×C22⋊C8—1221no—
273C2×C22⋊C8—1221no—
274C2×C4×C8—612no—
275#336—1221no—
276#336—1221no—
277Q8⋊C8—1224no—
278Q8⋊C8—1224no—
279C61#831111yescharacteristic, complemented, Sylow
280C22×Q8—614no—
281C22×Q8—611no—
282C22×Q8—1221no—
283C22×Q8—1222no—
284C22×Q8—1222no—
285C22×Q8—1222no—
286C22×Q8—1222no—
287C22×Q8—1222no—
288C22×Q8C2×#692311yescharacteristic
289C23×C4—611no—
290C23×C4—611no—
291C23×C4—611no—
292C23×C4—1221no—
293C23×C4—1221no—
294C22×C8—1222no—
295C22×C8—1222no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—614no—
299C4×Q8—614no—
300C4×Q8—1222no—
301C4×Q8—1222no—
302C4×Q8—1228no—
303C4×Q8—1222no—
304C4×Q8—1222no—
305C4×Q8—1228no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1228no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4C2×#692311yescharacteristic
316C2×C42—21no—
317C2×C42—611no—
318C2×C42—611no—
319C2×C42—611no—
320C2×C42—1222no—
321C2×C42—1222no—
322C2×C42—1228no—
323C2×C42—1221no—
324C2×C42—1221no—
325C2×C42—1221no—
326C2×C42C2×#692311yescharacteristic
327C4⋊C8—1222no—
328C4⋊C8—1222no—
329C22⋊C8—1224no—
330C22⋊C8—2442no—
331C4×C8—1222no—
332C4×C8—1222no—
333C24—611no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—612no—
338C2×Q8—612no—
339C2×Q8—612no—
340C2×Q8—1224no—
341C2×Q8—1224no—
342C2×Q8—1224no—
343C2×Q8—1228no—
344C2×Q8—1228no—
345C2×Q8—1228no—
346C2×Q8—1228no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×Q8D61⋊C812yes—
351C2×Q8#1175821312yes—
352C2×Q8C22⋊#692312yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—612no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—612no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—1224no—
366C22×C4—1224no—
367C22×C4—1224no—
368C22×C4—1222no—
369C22×C4—1224no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—2441no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4C22⋊#692311yescharacteristic
380C22×C4C22⋊#692311yescharacteristic
381C22×C4C22×#692311yescharacteristic
382C2×C8—1224no—
383C2×C8—2442no—
384C2×C8—2442no—
385C2×C8—2442no—
386C2×C8—2442no—
387C2×C8—2442no—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—612no—
391C4⋊C4—1224no—
392C4⋊C4—1224no—
393C4⋊C4—1224no—
394C4⋊C4—1224no—
395C4⋊C4—1221no—
396C4⋊C4C22⋊#692312yes—
397C42—24no—
398C42—21no—
399C42—21no—
400C42—612no—
401C42—612no—
402C42—1222no—
403C42—1222no—
404C42—1224no—
405C42—1224no—
406C42—1224no—
407C23—612no—
408C23—612no—
409C23—612no—
410C23—1221no—
411C23—1221no—
412C23—1221no—
413C23—1221no—
414C23C2×C22⋊#692311yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—22no—
418Q8—614no—
419Q8—12216no—
420Q8—1222no—
421Q8—1222no—
422Q8—1222no—
423Q8#1451983414yescomplemented
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—611no—
436C2×C4—611no—
437C2×C4—612no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—1224no—
442C2×C4—1224no—
443C2×C4—12216no—
444C2×C4—1222no—
445C2×C4—1221no—
446C2×C4—1222no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—2444no—
457C2×C4—2448no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4#1451984012yes—
461C2×C4C2×C22⋊#692311yescharacteristic
462C2×C4C2×C22⋊#692311yescharacteristic
463C2×C4C2×D61⋊C811yescharacteristic
464C2×C4D61⋊M1611yescharacteristic
465C2×C4D61⋊M1611yescharacteristic
466C2×C4C2×#1175821311yescharacteristic
467C8—4882no—
468C8—2444no—
469C22—614no—
470C22—1222no—
471C22—1222no—
472C22—1222no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22#1684070311yescharacteristic
477C22D61⋊(C22⋊C8)11yescharacteristic
478C22#1684073311yescharacteristic
479C22C2×#1451984011yescharacteristic
480C22#1684075511yescharacteristic
481C22D61⋊#10411yescharacteristic
482C22#1684075911yescharacteristic
483C4—24no—
484C4—22no—
485C4—41no—
486C4—42no—
487C4—612no—
488C4—612no—
489C4—1222no—
490C4—1224no—
491C4—1222no—
492C4—2442no—
493C4D61⋊(C22⋊C8)11yescharacteristic
494C4C2×#1451983411yescharacteristic
495C4#1684073311yescharacteristic
496C4#1684073411yescharacteristic
497C2—1224no—
498C2D61⋊(Q8⋊C8)11yescharacteristic
499C2#2146112211yescharacteristic
500C2#2146114811yescharacteristic
501C2#2146116211yescharacteristic
502C2#2146164211yescharacteristic
503C2#2146165811yescharacteristic
504C2#2146166111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8311
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
4. C2×Q8⋊#92420C2semidirect
42. C2×Q8×D61C8semidirect
165. Q8×C61C22⋊C8semidirect
213. Dic122C22⋊C8semidirect
230. C2×D61Q8⋊C8semidirect
279. C61#8311semidirect
423. Q8#14519834semidirect

Maximal subgroups

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

subgroupindex
#214616742
C4×#145198342
C2×#168406122
C2×Q8⋊#924202
C2×C4×Q8×D612
#831161