Finite Group Databasebeta

← #30602016

Subgroups — #30602016

Order 15616.

522 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461699C211yesmaximal, characteristic, complemented
2C8×C22⋊#6923C211yesmaximal, characteristic, complemented
3#21461218C211yesmaximal, characteristic, complemented
4#21461215C211yesmaximal, characteristic
5#21461097C211yesmaximal, characteristic, complemented
6#21461094C211yesmaximal, characteristic
7C2×#16839990C211yesmaximal, characteristic
8C2×#14519848—21no—
9#16840733C2211yescharacteristic
10#16840733C2211yescharacteristic
11D61⋊(C22⋊C8)C2211yescharacteristic
12C2×C8×#6923—21no—
13C4×C22⋊#6923C2211yescharacteristic
14#16840638—21no—
15(C2×#6923)⋊C8—21no—
16#16840615—22no—
17#16840613—22no—
18#16840607—22no—
19#16840605—22no—
20C22×#11758200C411yescharacteristic, complemented
21C22×C8×D61C2211yescharacteristic
22C2×#14519757C2211yescharacteristic
23C2×#14519748C2211yescharacteristic
24#16839990—21no—
25#16839990—21no—
26#16839990—21no—
27#16839990—21no—
28C2×#14519708C411yescharacteristic, complemented
29C2×#14519702C411yescharacteristic, complemented
30C2×#102×C61C411yescharacteristic, complemented
31C2×#11758213—21no—
32C2×#11758213—21no—
33C2×D61⋊C8—21no—
34C2×C22⋊#6923—21no—
35C2×C4×#6923—21no—
36#14519848—42no—
37#14519834—21no—
38#14519834—21no—
39#14519834—21no—
40C22⋊#92420—21no—
41C22⋊#92420—21no—
42C22⋊#92420D412yes—
43C8×#6923—42no—
44(C2×#6923)⋊C4D412yes—
45C22×Dic122C2×C411yescharacteristic
46C22×C22⋊D61—21no—
47C22×C4×D61C2311yescharacteristic
48C2×#11758200—21no—
49C2×#11758200—21no—
50C2×#11758200—42no—
51C22×#92418C2×C411yescharacteristic
52C2×C8×D61—42no—
53C2×C8×D61D411yescharacteristic
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56C2×C8×D61D411yescharacteristic
57#14519757—21no—
58#14519757—21no—
59#14519757—21no—
60#14519757—21no—
61#14519748—21no—
62#14519748—21no—
63#14519748—21no—
64#14519748—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×C22⋊Dic61—21no—
68C2×#11758198—21no—
69#14519708—21no—
70#14519708—21no—
71#14519702—21no—
72#14519702—21no—
73C22×D4×C61C2×C411yescharacteristic
74C22×C8×C61C2×C411yescharacteristic
75C2×C4⋊C4×C61C2×C411yescharacteristic
76#102×C61—21no—
77#102×C61—21no—
78C24×C61—21no—
79C23×D61#4111yescharacteristic
80C22×#6923—22no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—82no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61C2×D411yescharacteristic
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61C2×D411yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×C4×D61C2×D411yescharacteristic
98C2×C4×D61#4111yescharacteristic
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C22⋊#6923—24no—
104C2×#92420—22no—
105C2×#92420—41no—
106C2×#92420—41no—
107C4×#6923—41no—
108C4×#6923—41no—
109#11758213—41no—
110#11758213—41no—
111#11758213—41no—
112#11758213—41no—
113D61⋊C8—41no—
114D61⋊C8—41no—
115C2×C8×C61—41no—
116C2×C8×C61C22⋊C411yescharacteristic
117C2×C8×C61C22⋊C411yescharacteristic
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122#11758200—42no—
123#11758200—42no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C2×#92418—22no—
131C2×#92418C22⋊C411yescharacteristic
132C2×#92418C22⋊C411yescharacteristic
133C8×D61—22no—
134C8×D61—22no—
135C8×D61—41no—
136C8×D61—41no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61—81no—
140C23×C61—81no—
141C23×C61C4×D411yescharacteristic
142C22×D61—22no—
143C22×D61—22no—
144C22×D61—41no—
145C22×D61#12111yescharacteristic
146C22×D61#12411yescharacteristic
147C2×#6923—24no—
148C2×#6923—42no—
149C2×#6923—42no—
150D4×C61—41no—
151D4×C61—41no—
152D4×C61—81no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C22⋊D61—82no—
160C22⋊D61—82no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—81no—
165C2×Dic61—81no—
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12711yescharacteristic
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—41no—
175Dic122—41no—
176Dic122—81no—
177#92420—42no—
178#92420—81no—
179#92420—81no—
180C8×C61—21no—
181C8×C61—21no—
182#92418—21no—
183#92418#9912yes—
184#9813—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—162no—
191C22×C61#39011yescharacteristic
192C22×C61#44312yes—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—44no—
199C2×D61#47112yes—
200#6923—44no—
201C4×C61—21no—
202C4×C61—81no—
203C4×C61C2×#9911yescharacteristic
204C4×C61#41011yescharacteristic
205Dic61—21no—
206Dic61—21no—
207Dic61—81no—
208Dic61—81no—
209C2×#465—611no—
210#1810—611no—
211C8×C22⋊C4—611no—
212#1508—611no—
213#1507—611no—
214#1472—611no—
215#1471—611no—
216C2×C61—22no—
217C2×C61—161no—
218C2×C61#196411yescharacteristic
219C2×C61#197411yescharacteristic
220C2×C61#197711yescharacteristic
221D61—22no—
222D61—41no—
223C22×SD16—611no—
224C23×C8—611no—
225C2×C22⋊Q8—611no—
226C2×#121—611no—
227#465—1221no—
228#465—1221no—
229#465—1221no—
230#465—1221no—
231C2×#108—1221no—
232C2×#106—611no—
233C2×#103—611no—
234C2×#102#692311yescharacteristic, complemented
235#407—611no—
236#407—611no—
237C2×C22⋊C8—611no—
238C2×C4×C8—1221no—
239C4×C22⋊C4—611no—
240#341—1221no—
241#336—1221no—
242#328—1222no—
243#327—1222no—
244#324—1222no—
245#323—1222no—
246C61#981311yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4—1221no—
249C22×D4C2×#692311yescharacteristic
250C23×C4—611no—
251C2×SD16—1221no—
252C2×SD16—1221no—
253C2×SD16—2442no—
254C2×M16—1221no—
255C2×M16—1221no—
256C22×C8—611no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C22×C8—611no—
261C22×C8—1221no—
262C22×C8—2442no—
263C22×C8C2×#692311yescharacteristic
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268#121—1221no—
269#121—1221no—
270#121—1221no—
271#121—1221no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4C2×#692311yescharacteristic
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C42—1221no—
279#108—2442no—
280#106—1221no—
281#106—1221no—
282#103—1221no—
283#103—1221no—
284#102—21no—
285#102—21no—
286C22⋊C8—612no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C4×C8—2442no—
293#95—612no—
294C24—21no—
295C24—611no—
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—4882no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—21no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—611no—
313C22×C4—1222no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4C22×#692311yescharacteristic
319SD16—2442no—
320SD16—2442no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324M16—2441no—
325C2×C8—41no—
326C2×C8—611no—
327C2×C8—611no—
328C2×C8—1222no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—1222no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8C22⋊#692311yescharacteristic
339C2×C8C22⋊#692311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—1224no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—2441no—
352C42—2441no—
353C23—41no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—611no—
358C23—611no—
359C23—1222no—
360C23—1222no—
361C23—2441no—
362C23D4×#692311yescharacteristic
363Q8—4881no—
364Q8—2441no—
365Q8—2441no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—4882no—
370D4—4882no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—611no—
376C2×C4—611no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1224no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4C2×C22⋊#692311yescharacteristic
392C2×C4D4×#692311yescharacteristic
393C8—21no—
394C8—21no—
395C8—612no—
396C8—1221no—
397C8—4881no—
398C8—4881no—
399C8—2442no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—162no—
406C22—612no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—1221no—
412C22—2444no—
413C22#1684071411yescharacteristic
414C22#1684080412yes—
415C4—21no—
416C4—81no—
417C4—1221no—
418C4—1221no—
419C4—4881no—
420C4—4881no—
421C4—2444no—
422C4D61⋊#9911yescharacteristic
423C4#1684074111yescharacteristic
424C2—22no—
425C2—161no—
426C2—1222no—
427C2—2441no—
428C2#2146196011yescharacteristic
429C2#2146197211yescharacteristic
430C2#2146197911yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#102×C61C4
socleC22×C61#390
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenterC2×#102#6923
hypocenterC61#9813
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #21461699C2semidirect
2. C8×C22⋊#6923C2semidirect
3. #21461218C2semidirect
5. #21461097C2semidirect
20. C22×#11758200C4semidirect
28. C2×#14519708C4semidirect
29. C2×#14519702C4semidirect
30. C2×#102×C61C4semidirect
234. C2×#102#6923semidirect
246. C61#9813semidirect

Maximal subgroups

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

subgroupindex
#214616992
C8×C22⋊#69232
#214612182
#214612152
#214610972
#214610942
C2×#168399902
#981361