Finite Group Databasebeta

← #30620325

Subgroups — #30620325

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462327C212yesmaximal, complemented
2#21462009C212yesmaximal, complemented
3#21461842C214yesmaximal
4#21461716C212yesmaximal, complemented
5#21461679C211yesmaximal, characteristic
6C4×#14519835C211yesmaximal, characteristic
7#21461518C212yesmaximal, complemented
8C2×#16840451C211yesmaximal, characteristic
9C2×#14519856C2214yes—
10C2×#14519835C2212yes—
11C2×#14519835C2212yes—
12#16840713—22no—
13#16840711—24no—
14#16840705C2214yes—
15#16840704C2212yes—
16C4⋊C4×#6923—22no—
17C4×C22⋊#6923C2212yes—
18#16840640C2211yescharacteristic
19#16840640C2214yes—
20#16840640C2212yes—
21#16840640C2211yescharacteristic
22#16840640C2214yes—
23#16840451—24no—
24#16840451—22no—
25#16840451—22no—
26C2×#14519770C412yescomplemented
27C2×#14519769C412yescomplemented
28C2×#14519768C411yescharacteristic, complemented
29C2×#14519759C2212yes—
30C2×#14519757C2211yescharacteristic
31C2×#14519756C411yescharacteristic, complemented
32C2×#14519742C411yescharacteristic, complemented
33C2×C4×C22⋊D61C2211yescharacteristic
34C2×#14519734C2212yes—
35C2×C4⋊C4×D61C2211yescharacteristic
36C2×#117×C61C411yescharacteristic, complemented
37#14519856—28no—
38#14519856—44no—
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—24no—
41C2×C22⋊#6923C2314yes—
42C2×C22⋊#6923C2312yes—
43C2×C22⋊#6923C2312yes—
44C2×C4×#6923—24no—
45C2×C4×#6923—22no—
46#14519835—24no—
47#14519835—42no—
48(C2×#6923)⋊C4—22no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—22no—
52C22×Dic122C2×C411yescharacteristic
53C22×C22⋊D61C2311yescharacteristic
54C22×C4×D61C2312yes—
55C22×C4×D61C2311yescharacteristic
56#14519770—24no—
57#14519770—24no—
58#14519769—24no—
59#14519769—24no—
60#14519768—22no—
61#14519768—22no—
62#14519759—24no—
63#14519759—24no—
64#14519757—22no—
65#14519757—22no—
66#14519756—22no—
67#14519756—22no—
68#14519742—22no—
69#14519742—21no—
70#14519742—21no—
71C4×C22⋊D61—22no—
72C4×C22⋊D61—22no—
73#14519734—24no—
74#14519734—24no—
75C2×C4⋊Dic61C2×C412yes—
76C2×C4⋊Dic61C2×C412yes—
77C2×C4⋊Dic61C2×C411yescharacteristic
78C2×#11758196C2×C412yes—
79C2×#11758196C2×C412yes—
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C412yes—
82C2×#11758196C2×C412yes—
83C4⋊C4×D61D414yes—
84C4⋊C4×D61D412yes—
85C4⋊C4×D61D412yes—
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C412yes—
88C2×#11758198C2311yescharacteristic
89C2×#11758198C2312yes—
90C2×C4×Dic61C2×C412yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C23×C4×C61C2×C411yescharacteristic
93#117×C61—22no—
94#117×C61—21no—
95#117×C61—21no—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C4⋊C4×C61C2×C411yescharacteristic
98C2×C22⋊C4×C61C2×C412yes—
99C2×C42×C61C2×C412yes—
100C24×C61C22×C411yescharacteristic
101C23×D61C2411yescharacteristic
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—24no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—42no—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C412yes—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C412yes—
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—21no—
115C2×C22⋊D61—21no—
116C22×Dic61C22×C411yescharacteristic
117C22×Dic61C22×C412yes—
118C22×Dic61C22×C412yes—
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C411yescharacteristic
121C2×C4×D61—24no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—42no—
125C2×C4×D61C2×D414yes—
126C2×C4×D61C2×D414yes—
127C2×C4×D61C2×D412yes—
128C2×C4×D61C2×D412yes—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—42no—
132C22⋊#6923—22no—
133C22⋊#6923—22no—
134C22⋊#6923—24no—
135C22⋊#6923—24no—
136C22⋊#6923—22no—
137C22⋊#6923#4114yes—
138C4⋊#6923—48no—
139C4⋊#6923—44no—
140C4×#6923—48no—
141C4×#6923—44no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61C22⋊C412yes—
145C4⋊C4×C61C22⋊C411yescharacteristic
146C4⋊C4×C61C22⋊C411yescharacteristic
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C42×C61—22no—
150C42×C61—22no—
151C22⋊Dic61—22no—
152C22⋊Dic61—24no—
153#11758198—22no—
154#11758198—24no—
155C4⋊Dic61—24no—
156C4⋊Dic61—24no—
157C4⋊Dic61C22⋊C412yes—
158C4⋊Dic61C22⋊C412yes—
159#11758196—22no—
160#11758196—22no—
161#11758196—24no—
162#11758196—24no—
163#11758196—22no—
164#11758196C22⋊C414yes—
165#11758196C22⋊C414yes—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61C23×C411yescharacteristic
176C22×D61—41no—
177C22×D61—41no—
178C22×D61C22×D412yes—
179C22×D61C2×#4112yes—
180C22×D61H(5,𝔽2)11yescharacteristic
181C22×D612−1+411yescharacteristic
182C2×#6923—24no—
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—42no—
186C2×#6923—44no—
187C2×#6923—42no—
188C2×#6923—44no—
189C2×#6923—44no—
190C2×#6923—42no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C22⋊D61—42no—
205C22⋊D61—42no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61C2×C22⋊C414yes—
217C2×Dic61C2×C22⋊C412yes—
218C4×D61—24no—
219C4×D61—24no—
220C4×D61—24no—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—42no—
224Dic122—42no—
225Dic122—42no—
226#22478—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61C22×C22⋊C411yescharacteristic
235C22×C61C4×#4111yescharacteristic
236C22×C61#52011yescharacteristic
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—42no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61#54612yes—
243C2×D61#54712yes—
244#6923—44no—
245#6923—84no—
246#6923—82no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61#41012yes—
253Dic61—24no—
254Dic61—22no—
255Dic61—41no—
256Dic61—42no—
257Dic61—41no—
258C2×#552—611no—
259#2344—612no—
260#1996—612no—
261#1899—614no—
262#1820—612no—
263#1799—611no—
264C4×#99—611no—
265#1740—612no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—81no—
269C2×C61#233911yescharacteristic
270C2×C61C2×#41011yescharacteristic
271C2×C61#288411yescharacteristic
272D61—22no—
273D61—22no—
274#552—1222no—
275#552—1224no—
276#552—1222no—
277C2×#126—612no—
278C2×#125—611no—
279C2×#125—612no—
280C2×#123—612no—
281C2×C22⋊Q8—611no—
282C2×C22⋊Q8—611no—
283C2×C4×Q8—611no—
284C2×C4×D4—611no—
285C2×#117—612no—
286C2×#117—614no—
287C2×#117#692311yescharacteristic, complemented
288C22×C4⋊C4—611no—
289C2×#99—612no—
290C2×#99—612no—
291#389—1222no—
292#387—1224no—
293#381—614no—
294#380—612no—
295C4×C4⋊C4—1222no—
296C4×C22⋊C4—612no—
297#342—612no—
298#342—611no—
299#342—614no—
300#342—614no—
301#342—611no—
302C61#2247811yescharacteristic, complemented, Sylow
303C22×Q8—611no—
304C22×D4—611no—
305C23×C4—612no—
306C23×C4—611no—
307C23×C4C2×#692311yescharacteristic
308#126—1224no—
309#126—1224no—
310#125—1224no—
311#125—1224no—
312#125—1222no—
313#125—1222no—
314#123—1224no—
315#123—1224no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C4×Q8—1222no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×D4—1222no—
324C4×D4—1222no—
325#117—22no—
326#117—21no—
327#117—21no—
328#117—1224no—
329#117—1224no—
330#117—1228no—
331#117—2444no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—614no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C2×#692311yescharacteristic
346C2×C4⋊C4C2×#692311yescharacteristic
347C2×C22⋊C4—612no—
348C2×C22⋊C4—611no—
349C2×C22⋊C4—612no—
350C2×C22⋊C4—611no—
351C2×C22⋊C4—612no—
352C2×C22⋊C4—614no—
353C2×C22⋊C4—612no—
354C2×C22⋊C4C2×#692312yes—
355C2×C42—612no—
356C2×C42—611no—
357C2×C42—1224no—
358C2×C42—1222no—
359C2×C42C2×#692312yes—
360#99—1224no—
361#99—2442no—
362#95—1222no—
363#95—1224no—
364#95—1222no—
365#95—1224no—
366C24—611no—
367C24C22×#692311yescharacteristic
368C2×Q8—1221no—
369C2×Q8—1221no—
370C2×Q8—2442no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—1221no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—612no—
379C22×C4—611no—
380C22×C4—614no—
381C22×C4—614no—
382C22×C4—612no—
383C22×C4—612no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—612no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1224no—
393C22×C4—2442no—
394C22×C4C22×#692312yes—
395C22×C4C22×#692311yescharacteristic
396C22×C4C22×#692311yescharacteristic
397C22×C4C22×#692312yes—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—614no—
401C4⋊C4—614no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1222no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—2448no—
412C4⋊C4—2444no—
413C4⋊C4C22⋊#692312yes—
414C4⋊C4C22⋊#692311yescharacteristic
415C4⋊C4C22⋊#692311yescharacteristic
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—614no—
419C22⋊C4—1222no—
420C22⋊C4—1224no—
421C22⋊C4—1222no—
422C22⋊C4—1224no—
423C22⋊C4—1222no—
424C22⋊C4—1222no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1222no—
428C42—22no—
429C42—22no—
430C42—1222no—
431C42—1222no—
432C42—1221no—
433C42—1221no—
434C42—2448no—
435C42—2444no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—612no—
442C23—612no—
443C23—611no—
444C23—611no—
445C23—2441no—
446C23—2441no—
447C23C23×#692311yescharacteristic
448Q8—2442no—
449Q8—2442no—
450D4—2442no—
451D4—2442no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—614no—
462C2×C4—612no—
463C2×C4—1224no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1224no—
470C2×C4—1224no—
471C2×C4—1221no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2444no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2444no—
482C2×C4—2442no—
483C2×C4—2444no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2441no—
487C2×C4—2442no—
488C2×C4C2×C22⋊#692312yes—
489C2×C4C2×C22⋊#692312yes—
490C2×C4C2×C22⋊#692311yescharacteristic
491C2×C4C2×C22⋊#692311yescharacteristic
492C22—21no—
493C22—21no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—612no—
500C22—612no—
501C22—1222no—
502C22—1222no—
503C22—2442no—
504C22—2441no—
505C22—2441no—
506C22C22×C22⋊#692311yescharacteristic
507C22#41×#692311yescharacteristic
508C22#1684083111yescharacteristic
509C4—22no—
510C4—22no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—1224no—
515C4—1222no—
516C4—4884no—
517C4—4882no—
518C4—2444no—
519C4—2441no—
520C4—2442no—
521C4—2441no—
522C4#1684074112yes—
523C2—21no—
524C2—21no—
525C2—81no—
526C2—1222no—
527C2—1222no—
528C2#2146232211yescharacteristic
529C2C2×#1684074111yescharacteristic
530C2#2146242111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC23×D61C24
hypercenterC2×#117#6923
hypocenterC61#22478
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21462327C2semidirect
2. #21462009C2semidirect
4. #21461716C2semidirect
7. #21461518C2semidirect
26. C2×#14519770C4semidirect
27. C2×#14519769C4semidirect
28. C2×#14519768C4semidirect
31. C2×#14519756C4semidirect
32. C2×#14519742C4semidirect
36. C2×#117×C61C4semidirect
287. C2×#117#6923semidirect
302. C61#22478semidirect

Maximal subgroups

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

subgroupindex
#214623272
#214620092
#214618422
#214617162
#214616792
C4×#145198352
#214615182
C2×#168404512
#2247861