Finite Group Databasebeta

← #30599880

Subgroups — #30599880

Order 15616.

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

normal subgroups only

This group has 488 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
2#21461989C211yesmaximal, characteristic, complemented
3#21461123C211yesmaximal, characteristic, complemented
4#21461112C211yesmaximal, characteristic
5#21460665C211yesmaximal, characteristic, complemented
6#21460652C211yesmaximal, characteristic, complemented
7#21454637C211yesmaximal, characteristic
8#16840812—21no—
9#16840796—21no—
10#16840791—22no—
11#16840784—21no—
12#16840778C2211yescharacteristic
13#16840778C2211yescharacteristic
14#16840766—21no—
15C2×C4⋊#92420C2211yescharacteristic
16C2×C4×#92420C2211yescharacteristic
17#16840628—22no—
18#16840612—21no—
19#16840612—21no—
20#16840612—21no—
21#16840610—21no—
22C2×#14519810C411yescharacteristic
23#16840230C2211yescharacteristic
24C2×#14519742C2211yescharacteristic
25#16839868—21no—
26#16839861—21no—
27#16839856—21no—
28#16839849—21no—
29#16839770C411yescharacteristic
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—
43C22⋊#92420—21no—
44C22⋊#92420—21no—
45#14519832—22no—
46C4×#92420—21no—
47C4×#92420—21no—
48C4×#92420—21no—
49C4×#92420—21no—
50C2×#11758222C2×C411yescharacteristic
51C22×Dic122C2×C411yescharacteristic
52#14519810—22no—
53#14519810—22no—
54C2×Dic244—21no—
55C2×#11758190—21no—
56C2×#11758194C2×C411yescharacteristic
57#14519742—21no—
58#14519742—21no—
59#14519742—21no—
60#14519742—21no—
61#14519742D411yescharacteristic
62#14519742D411yescharacteristic
63#14519739D411yescharacteristic
64#14519739D411yescharacteristic
65#14519737—21no—
66#14519737—21no—
67#14519734—21no—
68C2×#11758196—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61C2311yescharacteristic
71C42×D61—21no—
72#14519700—21no—
73#14519700—21no—
74#14519700—21no—
75#14519697—21no—
76#14519687—21no—
77C8×Dic61—21no—
78C2×M16×C61C2×C411yescharacteristic
79#117×C61C2×C411yescharacteristic
80C2×C4⋊C4×C61C2×C411yescharacteristic
81#107×C61—21no—
82#106×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61C22×C411yescharacteristic
85C2×C22⋊D61—21no—
86C22×Dic61—21no—
87C22×Dic61C2×D411yescharacteristic
88C22×Dic61#4111yescharacteristic
89#11758222—21no—
90#11758222—21no—
91#11758222—21no—
92#11758222—21no—
93C2×D244C22⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122C22⋊C411yescharacteristic
103C2×Dic122M1612yescomplemented
104C2×#92420—22no—
105C2×#92420—21no—
106C2×#92420—21no—
107C2×#92420—22no—
108C2×#92420—41no—
109C2×#92420—41no—
110C2×#92420—42no—
111C2×#92420—41no—
112#11758213—82no—
113M16×C61—21no—
114M16×C61M1612yescomplemented
115C2×C8×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61C22⋊C411yescharacteristic
119C4⋊C4×C61C22⋊C411yescharacteristic
120C22⋊C4×C61—21no—
121C42×C61—21no—
122#11758198—21no—
123#11758198—21no—
124#11758196—21no—
125#11758196—21no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C4×Dic61C2×D411yescharacteristic
133C4×Dic61C2×D411yescharacteristic
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61#4111yescharacteristic
136#11758194—21no—
137#11758194—21no—
138C2×#92418—21no—
139Dic244—24no—
140#11758190—24no—
141C23×C61C4×D411yescharacteristic
142C22×D61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C2×M1611yescharacteristic
150C2×C4×C61C2×M1611yescharacteristic
151C2×C4×C61#13111yescharacteristic
152C2×C4×C61#13111yescharacteristic
153C22⋊D61—41no—
154C22⋊D61—41no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—22no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61#12011yescharacteristic
163C2×Dic61#12111yescharacteristic
164C2×Dic61#12111yescharacteristic
165C2×Dic61#12311yescharacteristic
166D244—21no—
167D244—21no—
168C4×D61—41no—
169C4×D61—41no—
170C4×D61—41no—
171C4×D61—41no—
172C4×D61—81no—
173Dic122—22no—
174Dic122—21no—
175Dic122—21no—
176Dic122—22no—
177Dic122—41no—
178Dic122—41no—
179#92420—42no—
180#92420—41no—
181#92420—42no—
182#92420—41no—
183#92420—81no—
184C8×C61—22no—
185#92418—41no—
186#8955—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#38611yescharacteristic
190C22×C61#43511yescharacteristic
191C22×C61#43611yescharacteristic
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—41no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61—81no—
198C4×C61#40711yescharacteristic
199C4×C61#40811yescharacteristic
200C4×C61#43511yescharacteristic
201C4×C61#43611yescharacteristic
202Dic61—21no—
203Dic61—21no—
204Dic61—21no—
205Dic61—42no—
206Dic61—41no—
207Dic61#45511yescharacteristic
208Dic61#45611yescharacteristic
209#2989—611no—
210C4×#137—611no—
211#1984—611no—
212#1479—611no—
213#1476—611no—
214#1283—611no—
215#1278—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61#187011yescharacteristic
219C2×C61#188111yescharacteristic
220C2×C61#188211yescharacteristic
221D61—81no—
222C2×#137—611no—
223C4×#41—611no—
224C2×C4×Q8—611no—
225#446—1221no—
226#441—1221no—
227#440—1221no—
228C4×Q16—1221no—
229C4×SD16—1221no—
230#436—1221no—
231#435—1222no—
232#432—1221no—
233#431—611no—
234#431—611no—
235#428#692311yescharacteristic
236#423—1221no—
237C2×C4⋊C8—611no—
238#419—611no—
239#417—611no—
240C4×M16—611no—
241C2×C4×C8—611no—
242#336—1222no—
243Q8⋊C8—1221no—
244Q8⋊C8—1221no—
245Q8⋊C8—1221no—
246D4⋊C8—1221no—
247C61#895511yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×Q8—611no—
250#137—1222no—
251#137—1222no—
252C2×Q16—1221no—
253C2×SD16—1221no—
254C2×M16—611no—
255C2×M16—1222no—
256C2×M16C2×#692311yescharacteristic
257C22×C8—612no—
258C22×C8—1221no—
259C4×Q8—611no—
260C4×Q8—611no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×Q8—1221no—
264C4×Q8—1221no—
265C4×D4—611no—
266C4×D4—611no—
267C4×D4—1221no—
268C4×D4—1221no—
269#117—1221no—
270#117C2×#692311yescharacteristic
271C2×C4⋊C4—1221no—
272C2×C4⋊C4C2×#692311yescharacteristic
273C2×C42—611no—
274C2×C42—1221no—
275C2×C42—1221no—
276#107—21no—
277#106—21no—
278C4⋊C8—611no—
279C4⋊C8—611no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1222no—
284#103—1221no—
285#103—1221no—
286#103—1221no—
287#102—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1222no—
290C22⋊C8—1221no—
291#97—1221no—
292#97—1222no—
293C4×C8—1221no—
294C4×C8—1221no—
295C4×C8—1221no—
296C4×C8—1221no—
297C4×C8—1221no—
298#41—1221no—
299#41—1221no—
300#41—1221no—
301#41—1221no—
302C2×Q8—611no—
303C2×Q8—612no—
304C2×Q8—1221no—
305C2×Q8—1221no—
306C2×Q8—1221no—
307C2×Q8—1221no—
308C2×Q8—1221no—
309C2×D4—611no—
310C2×D4—1221no—
311C22×C4—21no—
312C22×C4—611no—
313C22×C4—611no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4C22×#692311yescharacteristic
319Q16—1224no—
320SD16—1224no—
321M16—21no—
322M16—1221no—
323M16—1221no—
324M16—4882no—
325M16#1175821312yescomplemented
326C2×C8—21no—
327C2×C8—1221no—
328C2×C8—1221no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—1221no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2442no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C4⋊C4C22⋊#692311yescharacteristic
343C4⋊C4C22⋊#692311yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C42—21no—
348C42—611no—
349C42—611no—
350C42—611no—
351C42—611no—
352C42—1221no—
353C42—1221no—
354C42—2441no—
355C42—2441no—
356C23—1221no—
357C23D4×#692311yescharacteristic
358Q8—1222no—
359Q8—1221no—
360Q8—1221no—
361Q8—1222no—
362Q8—2441no—
363Q8—2441no—
364D4—1221no—
365D4—1221no—
366D4—2441no—
367D4—2441no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1222no—
380C2×C4—4881no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4C2×C22⋊#692311yescharacteristic
389C2×C4D4×#692311yescharacteristic
390C2×C4D61⋊M1611yescharacteristic
391C2×C4C2×#1175821311yescharacteristic
392C2×C4#1451986511yescharacteristic
393C2×C4#1451986511yescharacteristic
394C8—22no—
395C8—4881no—
396C8—2441no—
397C8—2442no—
398C8—2441no—
399C8—2442no—
400C8—2441no—
401C22—21no—
402C22—21no—
403C22—2441no—
404C22—2441no—
405C22—2441no—
406C22#1684071011yescharacteristic
407C22#1684079211yescharacteristic
408C22#1684079611yescharacteristic
409C4—41no—
410C4—41no—
411C4—81no—
412C4—611no—
413C4—611no—
414C4—1221no—
415C4—1221no—
416C4—1221no—
417C4—2441no—
418C4—2442no—
419C4#1684073311yescharacteristic
420C4#1684073711yescharacteristic
421C4#1684079211yescharacteristic
422C4#1684079611yescharacteristic
423C2—21no—
424C2—21no—
425C2—4881no—
426C2#2146179911yescharacteristic
427C2#2146181711yescharacteristic
428C2#2146181811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#428C4
socleC22×C61#386
prime radicalC2×#11758222C2×C4
cosocleC2×C4×Dic61C23
hypercenter#428#6923
hypocenterC61#8955
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
2. #21461989C2semidirect
3. #21461123C2semidirect
5. #21460665C2semidirect
6. #21460652C2semidirect
103. C2×Dic122M16semidirect
114. M16×C61M16semidirect
247. C61#8955semidirect
325. M16#11758213semidirect

Maximal subgroups

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

subgroupindex
#214627402
#214619892
#214611232
#214611122
#214606652
#214606522
#214546372
#895561