Finite Group Databasebeta

← #30599884

Subgroups — #30599884

Order 15616.

540 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#21462740C211yesmaximal, characteristic, complemented
2#21461941C211yesmaximal, characteristic, complemented
3#21461120C211yesmaximal, characteristic, complemented
4#21461112C211yesmaximal, characteristic, complemented
5#21460665C211yesmaximal, characteristic, complemented
6#21460652C211yesmaximal, characteristic, complemented
7#21454626C211yesmaximal, characteristic
8#16840812—21no—
9#16840796—21no—
10#16840791—22no—
11#16840784—21no—
12#16840778C2211yescharacteristic
13#16840778C2211yescharacteristic
14#16840766—21no—
15C2×C4⋊#92420C2211yescharacteristic
16C2×#14519834—22no—
17C2×C4×#92420C2211yescharacteristic
18#16840612—21no—
19#16840610—21no—
20#16840610—21no—
21#16840610—21no—
22C2×#14519805C411yescharacteristic
23#16840230C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25#16839873—21no—
26#16839861—21no—
27#16839849—21no—
28#16839844—21no—
29#16839766C411yescharacteristic
30#16839761C411yescharacteristic
31M16×Dic61C2211yescharacteristic
32C61×#428C411yescharacteristic
33C2×#11758213—22no—
34C22×#92420—21no—
35C22×#92420D412yes—
36C4⋊#92420—22no—
37C4⋊#92420—21no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40C4⋊#92420D411yescharacteristic
41C4⋊#92420D411yescharacteristic
42#14519834—22no—
43#14519834—44no—
44C22⋊#92420—21no—
45C22⋊#92420—21no—
46#14519832—22no—
47C4×#92420—21no—
48C4×#92420—21no—
49C4×#92420—21no—
50C4×#92420—21no—
51C2×#11758222C2×C411yescharacteristic
52C22×D244C2×C411yescharacteristic
53C22×C4×D61—21no—
54#14519805—22no—
55#14519805—22no—
56C2×#11758190—21no—
57C2×D488—21no—
58C2×#11758194C2×C411yescharacteristic
59#14519742D411yescharacteristic
60#14519742D411yescharacteristic
61#14519739—21no—
62#14519739—21no—
63#14519739—21no—
64#14519739—21no—
65#14519739D411yescharacteristic
66#14519739D411yescharacteristic
67#14519737—21no—
68#14519737—21no—
69#14519734—21no—
70C2×#11758198—21no—
71C2×C4×Dic61C2311yescharacteristic
72C42×D61—21no—
73#14519700—21no—
74#14519697—21no—
75#14519697—21no—
76#14519697—21no—
77#14519687—21no—
78C8×Dic61—21no—
79C2×M16×C61C2×C411yescharacteristic
80#117×C61C2×C411yescharacteristic
81C2×C4⋊C4×C61C2×C411yescharacteristic
82#107×C61—21no—
83#106×C61—21no—
84C23×D61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—21no—
88C22×Dic61C2×D411yescharacteristic
89C22×Dic61#4111yescharacteristic
90#11758222—21no—
91#11758222—21no—
92#11758222—21no—
93#11758222—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244C22⋊C411yescharacteristic
100C2×D244M1612yescomplemented
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—42no—
105C2×C4×D61—42no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×Dic122C22⋊C411yescharacteristic
109C2×#92420—22no—
110C2×#92420—21no—
111C2×#92420—21no—
112C2×#92420—22no—
113C2×#92420—41no—
114C2×#92420—41no—
115C2×#92420—42no—
116C2×#92420—41no—
117#11758213—82no—
118M16×C61—21no—
119M16×C61M1612yescomplemented
120C2×C8×C61—21no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61C22⋊C411yescharacteristic
124C4⋊C4×C61C22⋊C411yescharacteristic
125C22⋊C4×C61—21no—
126C42×C61—21no—
127#11758198—21no—
128#11758198—21no—
129#11758198—41no—
130#11758198—41no—
131#11758196—21no—
132#11758196—21no—
133C4×Dic61—21no—
134C4×Dic61—21no—
135C4×Dic61C2×D411yescharacteristic
136C4×Dic61C2×D411yescharacteristic
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61#4111yescharacteristic
139#11758194—21no—
140#11758194—21no—
141C2×#92418—21no—
142D488—24no—
143#11758190—24no—
144C23×C61C4×D411yescharacteristic
145C22×D61—21no—
146C22×D61—22no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61—41no—
150C22×D61—41no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61C2×C22⋊C411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61C2×M1611yescharacteristic
158C2×C4×C61C2×M1611yescharacteristic
159C2×C4×C61#13111yescharacteristic
160C2×C4×C61#13111yescharacteristic
161C22⋊D61—41no—
162C22⋊D61—41no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61#12011yescharacteristic
168C2×Dic61#12111yescharacteristic
169C2×Dic61#12111yescharacteristic
170C2×Dic61#12311yescharacteristic
171D244—22no—
172D244—21no—
173D244—21no—
174D244—22no—
175D244—41no—
176D244—41no—
177C4×D61—41no—
178C4×D61—41no—
179C4×D61—41no—
180C4×D61—41no—
181C4×D61—81no—
182C4×D61—82no—
183C4×D61—81no—
184C4×D61—81no—
185Dic122—21no—
186Dic122—21no—
187#92420—42no—
188#92420—41no—
189#92420—42no—
190#92420—41no—
191#92420—81no—
192C8×C61—22no—
193#92418—41no—
194#8956—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61#38611yescharacteristic
198C22×C61#43511yescharacteristic
199C22×C61#43611yescharacteristic
200C2×D61—44no—
201C2×D61—41no—
202C2×D61—42no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61—82no—
206C2×D61—81no—
207C2×D61—81no—
208C4×C61—41no—
209C4×C61—41no—
210C4×C61—81no—
211C4×C61#40711yescharacteristic
212C4×C61#40811yescharacteristic
213C4×C61#43511yescharacteristic
214C4×C61#43611yescharacteristic
215Dic61—21no—
216Dic61—21no—
217Dic61—21no—
218Dic61—41no—
219Dic61#45311yescharacteristic
220Dic61#45411yescharacteristic
221#2989—611no—
222C4×AGL(1,ℤ/8)—611no—
223#1954—611no—
224#1478—611no—
225#1476—611no—
226#1283—611no—
227#1278—611no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61#187011yescharacteristic
231C2×C61#188111yescharacteristic
232C2×C61#188311yescharacteristic
233D61—81no—
234D61—82no—
235C2×AGL(1,ℤ/8)—611no—
236C4×#41—611no—
237C2×C4×D4—611no—
238#446—1221no—
239#442—1221no—
240#440—1221no—
241C4×SD16—1221no—
242C4×D8—1221no—
243#436—1221no—
244#435—1222no—
245#432—1221no—
246#431—611no—
247#431—611no—
248#428#692311yescharacteristic
249#423—1221no—
250C2×C4⋊C8—611no—
251#418—611no—
252#417—611no—
253C2×C22⋊C8—1222no—
254C4×M16—611no—
255C2×C4×C8—611no—
256Q8⋊C8—1221no—
257D4⋊C8—1221no—
258D4⋊C8—1221no—
259D4⋊C8—1221no—
260C61#895611yescharacteristic, complemented, Sylow
261C2×#41—611no—
262C22×D4—611no—
263C23×C4—1221no—
264AGL(1,ℤ/8)—1222no—
265AGL(1,ℤ/8)—1222no—
266C2×SD16—1221no—
267C2×D8—1221no—
268C2×M16—611no—
269C2×M16—1222no—
270C2×M16C2×#692311yescharacteristic
271C22×C8—612no—
272C22×C8—1221no—
273C4×Q8—611no—
274C4×Q8—611no—
275C4×D4—611no—
276C4×D4—611no—
277C4×D4—1221no—
278C4×D4—1221no—
279C4×D4—1221no—
280C4×D4—1221no—
281C4×D4—1221no—
282C4×D4—1221no—
283#117—1221no—
284#117C2×#692311yescharacteristic
285C2×C4⋊C4C2×#692311yescharacteristic
286C2×C22⋊C4—1221no—
287C2×C42—611no—
288C2×C42—1221no—
289#107—21no—
290#106—21no—
291C4⋊C8—611no—
292C4⋊C8—611no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1222no—
297#103—1221no—
298#102—1221no—
299#102—1221no—
300#102—1221no—
301C22⋊C8—1221no—
302C22⋊C8—1222no—
303C22⋊C8—1221no—
304C22⋊C8—2444no—
305#97—1221no—
306#97—1222no—
307C4×C8—1221no—
308C4×C8—1221no—
309C4×C8—1221no—
310C4×C8—1221no—
311C4×C8—1221no—
312C24—1221no—
313#41—1221no—
314#41—1221no—
315#41—1221no—
316#41—1221no—
317C2×Q8—611no—
318C2×D4—611no—
319C2×D4—612no—
320C2×D4—1221no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C22×C4—21no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4—2442no—
335C22×C4—2442no—
336C22×C4C22×#692311yescharacteristic
337SD16—1224no—
338D8—1224no—
339M16—21no—
340M16—1221no—
341M16—1221no—
342M16—4882no—
343M16#1175821312yescomplemented
344C2×C8—21no—
345C2×C8—1221no—
346C2×C8—1221no—
347C2×C8—1222no—
348C2×C8—1222no—
349C2×C8—1221no—
350C2×C8—2441no—
351C2×C8—2441no—
352C2×C8—2441no—
353C2×C8—2442no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4C22⋊#692311yescharacteristic
359C4⋊C4C22⋊#692311yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C22⋊C4—2441no—
364C22⋊C4—2441no—
365C42—21no—
366C42—611no—
367C42—611no—
368C42—611no—
369C42—611no—
370C42—1221no—
371C42—1221no—
372C23—1221no—
373C23—1222no—
374C23—2442no—
375C23—2442no—
376C23—2441no—
377C23—2441no—
378C23D4×#692311yescharacteristic
379Q8—1221no—
380Q8—1221no—
381D4—1221no—
382D4—1222no—
383D4—1221no—
384D4—1222no—
385D4—2441no—
386D4—2441no—
387D4—2441no—
388D4—2441no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—611no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—4881no—
401C2×C4—4881no—
402C2×C4—4881no—
403C2×C4—4882no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4C2×C22⋊#692311yescharacteristic
410C2×C4D4×#692311yescharacteristic
411C2×C4D61⋊M1611yescharacteristic
412C2×C4C2×#1175821311yescharacteristic
413C2×C4#1451986511yescharacteristic
414C2×C4#1451986511yescharacteristic
415C8—22no—
416C8—4881no—
417C8—2441no—
418C8—2442no—
419C8—2441no—
420C8—2442no—
421C8—2441no—
422C22—21no—
423C22—21no—
424C22—4882no—
425C22—4881no—
426C22—4881no—
427C22—2444no—
428C22—2441no—
429C22—2442no—
430C22—2441no—
431C22—2441no—
432C22#1684071011yescharacteristic
433C22#1684079211yescharacteristic
434C22#1684079611yescharacteristic
435C4—41no—
436C4—41no—
437C4—81no—
438C4—611no—
439C4—611no—
440C4—1221no—
441C4—1221no—
442C4—1221no—
443C4—2441no—
444C4#1684073311yescharacteristic
445C4#1684073711yescharacteristic
446C4#1684079211yescharacteristic
447C4#1684079611yescharacteristic
448C2—21no—
449C2—21no—
450C2—4882no—
451C2—4881no—
452C2#2146179911yescharacteristic
453C2#2146181611yescharacteristic
454C2#2146181911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#428C4
socleC22×C61#386
prime radicalC2×#14519805C4
cosocleC2×C4×Dic61C23
hypercenter#428#6923
hypocenterC61#8956
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #21462740C2semidirect
2. #21461941C2semidirect
3. #21461120C2semidirect
4. #21461112C2semidirect
5. #21460665C2semidirect
6. #21460652C2semidirect
100. C2×D244M16semidirect
119. M16×C61M16semidirect
260. C61#8956semidirect
343. M16#11758213semidirect

Maximal subgroups

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

subgroupindex
#214627402
#214619412
#214611202
#214611122
#214606652
#214606522
#214546262
#895661