Finite Group Databasebeta

← #29595504

Subgroups — #29595504

Order 15616.

1098 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C2×#16839556C212yesmaximal, complemented
3C2×#16839554C212yesmaximal, complemented
4#21445737C211yesmaximal, characteristic, complemented
5C61×#1782C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic, complemented
7#16840308—24no—
8#16840308—24no—
9C2×#14519775C411yescharacteristic, complemented
10C2×#14519773C2211yescharacteristic
11C2×#14519766C411yescharacteristic, complemented
12C2×#14519764C2211yescharacteristic
13C2×#14519749—21no—
14C2×#14519749—21no—
15C2×C4×C22⋊D61—21no—
16C2×C4×C22⋊D61—21no—
17C22×C4×Dic61C411yescharacteristic, complemented
18C2×#14519692—21no—
19C2×#14519692C2212yes—
20#16839625—21no—
21#16839556—24no—
22#16839554—24no—
23C22×#41×C61C2211yescharacteristic
24C2×#99×C61—21no—
25C2×#99×C61C2212yes—
26C61×#381—21no—
27C2×#11758222—22no—
28C2×#11758222—22no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122C2×C411yescharacteristic
32C22×D244C2×C411yescharacteristic
33C22×C22⋊D61—21no—
34C22×C22⋊D61—21no—
35C22×C22⋊D61—21no—
36C23×Dic61—21no—
37C22×C4×D61—21no—
38#14519775—22no—
39#14519775—22no—
40#14519773—22no—
41#14519773D414yes—
42#14519766—22no—
43#14519766—22no—
44#14519764—22no—
45#14519764D414yes—
46#14519749—28no—
47#14519749—28no—
48C4×C22⋊D61—28no—
49C4×C22⋊D61—28no—
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61—22no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61C2×C411yescharacteristic
61C2×C4×Dic61C2×C411yescharacteristic
62#14519692—22no—
63#14519692—22no—
64#14519692—42no—
65#14519684—42no—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—42no—
69C2×#41×C61—42no—
70C22×Q8×C61C2×C411yescharacteristic
71C22×D4×C61D411yescharacteristic
72C22×D4×C61D411yescharacteristic
73C22×D4×C61C2311yescharacteristic
74C23×C4×C61C2×C411yescharacteristic
75C23×C4×C61D411yescharacteristic
76C23×C4×C61D411yescharacteristic
77C2×C22⋊C4×C61—22no—
78#99×C61—22no—
79#99×C61—22no—
80#99×C61—42no—
81#95×C61—42no—
82C24×C61C22⋊C411yescharacteristic
83C24×C61C2×D411yescharacteristic
84C24×C61C2×D411yescharacteristic
85C23×D61—21no—
86#41×C61—48no—
87#41×C61—48no—
88C2×Q8×C61—21no—
89C2×Q8×C61—21no—
90C2×Q8×C61—41no—
91C2×Q8×C61—41no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61C2×D412yes—
101C2×D4×C61C2×D412yes—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61—42no—
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C411yescharacteristic
115C22×C4×C61C22⋊C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—44no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—24no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—44no—
129C22×Dic61—42no—
130#11758222—48no—
131#11758222—48no—
132C2×D244—21no—
133C2×D244—41no—
134C2×D244—41no—
135C2×D244C22⋊C412yes—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×Dic122—21no—
140C2×Dic122—41no—
141C2×Dic122—41no—
142C2×Dic122C22⋊C412yes—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—42no—
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147C22⋊Dic61—42no—
148C22⋊Dic61—42no—
149#11758198—24no—
150#11758198—24no—
151#11758196—24no—
152#11758196—24no—
153C4×Dic61—24no—
154C4×Dic61—21no—
155C4×Dic61—21no—
156C4×Dic61C22⋊C412yes—
157C4×Dic61C22⋊C412yes—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61#9912yes—
168C23×C61C2×C22⋊C411yescharacteristic
169C23×C61#12012yes—
170C23×C61#12012yes—
171C22×D61—22no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—41no—
175C22×D61—41no—
176Q8×C61—42no—
177Q8×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—44no—
181D4×C61—44no—
182C2×C4×C61—24no—
183C2×C4×C61—24no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—44no—
195C2×C4×C61—44no—
196C2×C4×C61—82no—
197C2×C4×C61—84no—
198C2×C4×C61#9912yes—
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C2×C22⋊C411yescharacteristic
201C22⋊D61—48no—
202C22⋊D61—48no—
203C22⋊D61—48no—
204C2×Dic61—24no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—48no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—82no—
213C2×Dic61—84no—
214D244—42no—
215D244—42no—
216C4×D61—48no—
217Dic122—42no—
218Dic122—42no—
219#12695—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61#37911yescharacteristic
234C22×C61C2×#9911yescharacteristic
235C22×C61C2×#9911yescharacteristic
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—82no—
240C2×D61—82no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—22no—
244C4×C61—41no—
245C4×C61—41no—
246C4×C61—162no—
247Dic61—24no—
248Dic61—42no—
249Dic61—42no—
250Dic61—162no—
251C2×#532—611no—
252C2×#354—612no—
253C2×#353—612no—
254#1782—611no—
255#1782D6111yescharacteristic, complemented
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—42no—
259C2×C61—42no—
260C2×C61—42no—
261C2×C61#178111yescharacteristic
262C2×C61#212612yes—
263D61—82no—
264C22×#41—611no—
265C22×#41C2×D6111yescharacteristic
266#532—1224no—
267#532—1224no—
268C2×C4⋊Q8—611no—
269C2×#127—611no—
270C2×#124—611no—
271C2×#124—611no—
272C2×#121—1221no—
273C2×#121—1221no—
274C2×C4×D4—1221no—
275C2×C4×D4—1221no—
276C22×C42—611no—
277C2×#99—21no—
278C2×#99—612no—
279C2×#99—1221no—
280C2×#99C2×D6112yes—
281#381—21no—
282#381—1221no—
283#354—1224no—
284#353—1224no—
285C61#1269511yescharacteristic, complemented, Sylow
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—42no—
289C2×#41—42no—
290C2×#41—1222no—
291C2×#41—1222no—
292C2×#41—2442no—
293C2×#41—2442no—
294C22×Q8—611no—
295C22×Q8C4×D6111yescharacteristic
296C22×D4—611no—
297C22×D4—1221no—
298C22×D4—1221no—
299C22×D4—1221no—
300C22×D4D24411yescharacteristic
301C22×D4C22⋊D6111yescharacteristic
302C22×D4C22×D6111yescharacteristic
303C23×C4—1221no—
304C23×C4—1221no—
305C23×C4C4×D6111yescharacteristic
306C23×C4D24411yescharacteristic
307C23×C4C22⋊D6111yescharacteristic
308C4⋊Q8—1222no—
309C4⋊Q8—1222no—
310#127—614no—
311#127—1222no—
312#124—614no—
313#124—1222no—
314#124—1222no—
315#124—1222no—
316#121—1228no—
317#121—1228no—
318C4×D4—1228no—
319C4×D4—1228no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1222no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—1222no—
331C2×C42—1222no—
332C2×C42—1221no—
333#99—22no—
334#99—22no—
335#99—42no—
336#99—1222no—
337#99—1222no—
338#99—2442no—
339#95—42no—
340#95—2442no—
341C24—1221no—
342C24#1175819811yescharacteristic
343C24C2×D24411yescharacteristic
344C24C2×C22⋊D6111yescharacteristic
345#41—48no—
346#41—48no—
347#41—2448no—
348#41—2448no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—612no—
354C2×Q8—1221no—
355C2×Q8—2441no—
356C2×Q8—2441no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—612no—
366C2×D4—1224no—
367C2×D4—1224no—
368C2×D4—1224no—
369C2×D4—1221no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2444no—
373C2×D4—2444no—
374C2×D4—2442no—
375C2×D4—2442no—
376C2×D4D4×D6112yes—
377C2×D4D4×D6112yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—1224no—
390C22×C4—1221no—
391C22×C4—1224no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—2442no—
395C22×C4—2444no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4#1175819811yescharacteristic
399C22×C4#1175819811yescharacteristic
400C22×C4#1175819811yescharacteristic
401C22×C4C2×C4×D6111yescharacteristic
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—1224no—
407C22⋊C4—1224no—
408C22⋊C4—1224no—
409C22⋊C4—1224no—
410C22⋊C4—2442no—
411C22⋊C4—2442no—
412C42—612no—
413C42—612no—
414C42—1224no—
415C42—1221no—
416C42—1221no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—1222no—
427C23—2442no—
428C23—2442no—
429C23—2441no—
430C23—2441no—
431C23#1451969112yes—
432C23C2×#1175819811yescharacteristic
433C23#1451974512yes—
434C23#1451974712yes—
435Q8—42no—
436Q8—42no—
437Q8—2442no—
438Q8—2442no—
439D4—42no—
440D4—42no—
441D4—44no—
442D4—44no—
443D4—2442no—
444D4—2442no—
445D4—2448no—
446D4—2448no—
447D4—2448no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—1224no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—4882no—
469C2×C4—4884no—
470C2×C4—2448no—
471C2×C4—2448no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4#1451969112yes—
476C2×C4C22⋊C4×D6111yescharacteristic
477C2×C4C22⋊C4×D6111yescharacteristic
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—4882no—
492C22—4882no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22#1683961711yescharacteristic
497C22C2×#1451969111yescharacteristic
498C22C2×#1451969111yescharacteristic
499C4—22no—
500C4—21no—
501C4—21no—
502C4—41no—
503C4—41no—
504C4—162no—
505C4—1224no—
506C4—9762no—
507C4—2442no—
508C4—2442no—
509C2—21no—
510C2—21no—
511C2—42no—
512C2—42no—
513C2—42no—
514C2—4882no—
515C2#2144573311yescharacteristic
516C2#2144748212yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1782C2
socleC22×C61C2×#99
prime radicalC2×#16840308C2
cosocleC22×D4×C61C23
hypercenter#1782D61
hypocenterC61#12695
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. C2×#16839556C2semidirect
3. C2×#16839554C2semidirect
4. #21445737C2semidirect
5. C61×#1782C2semidirect
6. C22×#11758222C4semidirect
9. C2×#14519775C4semidirect
11. C2×#14519766C4semidirect
17. C22×C4×Dic61C4semidirect
255. #1782D61semidirect
285. C61#12695semidirect

Maximal subgroups

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

subgroupindex
C2×#168403082
C2×#168395562
C2×#168395542
#214457372
C61×#17822
#1269561