Finite Group Databasebeta

← #30607719

Subgroups — #30607719

Order 15616.

702 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C22×#14519838C211yesmaximal, characteristic, complemented
2C2×#16840671C212yesmaximal, complemented
3#21461653C211yesmaximal, characteristic, complemented
4#21460916C212yesmaximal, complemented
5#21449477C211yesmaximal, characteristic
6C22×#11758217—21no—
7C2×#14519838—24no—
8C2×#14519838—21no—
9C2×#14519838—21no—
10C2×#14519838C2212yes—
11C2×#14519838C2212yes—
12#16840732—21no—
13#16840671—24no—
14C23×Dic122C2211yescharacteristic
15C2×#14519765C411yescharacteristic
16C2×#14519765C2211yescharacteristic
17C2×#14519751C411yescharacteristic
18#16839675—21no—
19#16839675—21no—
20#16839668C411yescharacteristic
21#16839668C2211yescharacteristic
22#16839639—21no—
23#16839639—21no—
24C4×C22⋊C4×C61C411yescharacteristic
25C2×#11758217—24no—
26C2×#11758217—22no—
27C2×#11758217—22no—
28C2×#11758217—41no—
29C2×#11758217—42no—
30C22×#92420—41no—
31#14519838—24no—
32#14519838—22no—
33#14519838—22no—
34#14519838—22no—
35#14519838—42no—
36#14519838D414yescomplemented
37C22⋊#92420—41no—
38C22⋊#92420—41no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—42no—
42C22×Dic122—41no—
43C22×Dic122—41no—
44C22×Dic122C2×C411yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×Dic122D411yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244C2×C411yescharacteristic
49C22×C22⋊D61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52#14519765—22no—
53#14519765—22no—
54#14519765—42no—
55#14519751—41no—
56#14519751—41no—
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66#14519684—21no—
67C23×C4×C61C2×C411yescharacteristic
68C2×C22⋊C4×C61—21no—
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2×C411yescharacteristic
71#95×C61—21no—
72C24×C61C22⋊C411yescharacteristic
73C23×D61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—41no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C2×C22⋊D61—44no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—22no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—41no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C22×Dic61C2×D411yescharacteristic
99C2×D244—22no—
100C2×D244—41no—
101C2×D244—41no—
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—22no—
105C2×Dic122—21no—
106C2×Dic122—22no—
107C2×Dic122—21no—
108C2×Dic122—41no—
109C2×Dic122—41no—
110C2×Dic122—41no—
111C2×Dic122—41no—
112C2×Dic122—41no—
113C2×Dic122—41no—
114C2×Dic122—44no—
115C2×Dic122—44no—
116C2×Dic122—44no—
117C2×Dic122—44no—
118C2×Dic122C2×D412yes—
119C2×Dic122#4112yes—
120#11758217—24no—
121#11758217—44no—
122#11758217—42no—
123#11758217—42no—
124#11758217—42no—
125C2×#92420—44no—
126C2×#92420—42no—
127C2×#92420—42no—
128C22⋊C4×C61—42no—
129C42×C61—22no—
130C42×C61—41no—
131#11758198—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—41no—
135#11758198—42no—
136#11758198—41no—
137C4⋊Dic61—41no—
138C23×C61—22no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—41no—
142C23×C61#9911yescharacteristic
143C23×C61#9911yescharacteristic
144C23×C61C2×C22⋊C411yescharacteristic
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61C4×D412yes—
164C22⋊D61—82no—
165C22⋊D61—82no—
166C2×Dic61—22no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61#9912yes—
184C2×Dic61#12011yescharacteristic
185C2×Dic61#12112yes—
186C2×Dic61#12311yescharacteristic
187D244—82no—
188Dic122—42no—
189Dic122—42no—
190Dic122—44no—
191Dic122—44no—
192Dic122—42no—
193Dic122—42no—
194#92420—44no—
195#92420—82no—
196#13092—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61#35512yescomplemented
207C22×C61#38611yescharacteristic
208C22×C61C2×#9911yescharacteristic
209C22×C61C2×#9911yescharacteristic
210C2×D61—82no—
211C2×D61—82no—
212C2×D61—81no—
213C2×D61—81no—
214C2×D61—81no—
215C4×C61—22no—
216C4×C61—41no—
217C4×C61—81no—
218C4×C61—81no—
219Dic61—21no—
220Dic61—22no—
221Dic61—22no—
222Dic61—21no—
223Dic61—41no—
224Dic61—81no—
225C22×#101—611no—
226#2436—611no—
227C2×#355—612no—
228#1784—611no—
229#1394—612no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—22no—
233C2×C61—41no—
234C2×C61#189811yescharacteristic
235C2×C61C2×#35511yescharacteristic
236C2×C61#213311yescharacteristic
237D61—161no—
238C23×Q8—611no—
239C22×M16—1221no—
240C2×#124—611no—
241C2×#124—611no—
242C2×#121—611no—
243C2×#101—612no—
244C2×#101—612no—
245C2×#101—1224no—
246C2×#101—1221no—
247C2×#101—1221no—
248#407—1221no—
249#393—1221no—
250#393—1221no—
251#391—611no—
252#391—611no—
253#386—1221no—
254#386—1221no—
255C4×C22⋊C4#692311yescharacteristic
256#355—1224no—
257C61#1309211yescharacteristic, complemented, Sylow
258C22×Q8—611no—
259C22×Q8—611no—
260C22×Q8—611no—
261C22×Q8—611no—
262C22×Q8—1222no—
263C22×Q8—1222no—
264C22×Q8—2441no—
265C22×Q8—2441no—
266C22×Q8—2442no—
267C22×D4—611no—
268C22×D4—1221no—
269C23×C4—611no—
270C23×C4—611no—
271C23×C4C2×#692311yescharacteristic
272C2×M16—1222no—
273C2×M16—1224no—
274C2×M16—1222no—
275C2×M16—2442no—
276C2×M16—2441no—
277C22×C8—2441no—
278#124—1222no—
279#124—1222no—
280#124—2442no—
281#121—2441no—
282#121—2441no—
283C2×C4⋊C4—611no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C42C2×#692311yescharacteristic
291C2×C42C2×#692311yescharacteristic
292#101—614no—
293#101—1222no—
294#101—1224no—
295#101—1222no—
296#101—1222no—
297#101—2442no—
298C22⋊C8—2441no—
299C22⋊C8—2441no—
300#95—21no—
301#95—1221no—
302#95—1221no—
303#95—1221no—
304#95—1221no—
305C24—1221no—
306C24C22⋊#692311yescharacteristic
307C2×Q8—612no—
308C2×Q8—612no—
309C2×Q8—1221no—
310C2×Q8—1222no—
311C2×Q8—1222no—
312C2×Q8—1222no—
313C2×Q8—1221no—
314C2×Q8—1222no—
315C2×Q8—2441no—
316C2×Q8—2441no—
317C2×Q8—2441no—
318C2×Q8—2441no—
319C2×Q8—2441no—
320C2×Q8—2441no—
321C2×Q8—2444no—
322C2×Q8—2444no—
323C2×Q8—2444no—
324C2×Q8—2444no—
325C2×D4—1222no—
326C2×D4—2441no—
327C2×D4—2441no—
328C2×D4—2444no—
329C2×D4—2441no—
330C2×D4—2441no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—611no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1221no—
342C22×C4—1222no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—2441no—
351C22×C4C22⋊#692311yescharacteristic
352C22×C4C22×#692311yescharacteristic
353M16—1224no—
354M16—2442no—
355M16—2442no—
356M16—2444no—
357M16—2442no—
358C2×C8—2442no—
359C2×C8—2444no—
360C2×C8—2442no—
361C4⋊C4—2441no—
362C22⋊C4—42no—
363C22⋊C4—2442no—
364C22⋊C4—2441no—
365C22⋊C4—2442no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2442no—
369C42—22no—
370C42—41no—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—41no—
375C23—2442no—
376C23—2442no—
377C23—2441no—
378C23—2441no—
379C23—2441no—
380C23#1451983611yescharacteristic
381C23#1451983611yescharacteristic
382C23C2×C22⋊#692311yescharacteristic
383Q8—2442no—
384Q8—2442no—
385Q8—2444no—
386Q8—2444no—
387Q8—2442no—
388Q8—2442no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—611no—
406C2×C4—612no—
407C2×C4—611no—
408C2×C4—612no—
409C2×C4—1222no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4D4×#692312yes—
427C8—4882no—
428C8—2444no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—4882no—
439C22—4882no—
440C22—4881no—
441C22—4881no—
442C22—4881no—
443C22#1684067112yescomplemented
444C22#1684071011yescharacteristic
445C22D61⋊#9911yescharacteristic
446C22C2×#1451983611yescharacteristic
447C4—22no—
448C4—41no—
449C4—81no—
450C4—81no—
451C4—1222no—
452C4—1221no—
453C4—1222no—
454C4—1221no—
455C4—4881no—
456C4—2441no—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—41no—
461C2—9761no—
462C2#2146184111yescharacteristic
463C2C2×#1684067111yescharacteristic
464C2#2146211611yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC4×C22⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC2×#14519751C4
cosocleC22×Dic122C23
hypercenterC4×C22⋊C4#6923
hypocenterC61#13092
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14519838C2semidirect
2. C2×#16840671C2semidirect
3. #21461653C2semidirect
4. #21460916C2semidirect
36. #14519838D4semidirect
206. C22×C61#355semidirect
257. C61#13092semidirect
443. C22#16840671semidirect

Maximal subgroups

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

subgroupindex
C22×#145198382
C2×#168406712
#214616532
#214609162
#214494772
#1309261