Finite Group Databasebeta

← #30602029

Subgroups — #30602029

Order 15616.

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

normal subgroups only

This group has 484 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
2#21461555C211yesmaximal, characteristic
3#21461219C211yesmaximal, characteristic
4#21461218C211yesmaximal, characteristic
5#21461099C211yesmaximal, characteristic
6#21461097C211yesmaximal, characteristic
7C2×#16840002C211yesmaximal, characteristic
8C2×#14519848—21no—
9#16840733C2211yescharacteristic
10#16840733C2211yescharacteristic
11D61⋊(C22⋊C8)C2211yescharacteristic
12C2×#14519831—21no—
13C4×C22⋊#6923C2211yescharacteristic
14#16840638—21no—
15(C2×#6923)⋊C8—21no—
16#16840616—22no—
17#16840615—22no—
18#16840608—22no—
19#16840607—22no—
20C22×#11758202C411yescharacteristic, complemented
21C22×C8×D61C2211yescharacteristic
22C2×#14519757C2211yescharacteristic
23C2×#14519753C2211yescharacteristic
24#16840002—21no—
25#16840002—21no—
26#16840002—21no—
27#16840002—21no—
28C2×#14519710C411yescharacteristic, complemented
29C2×#14519702C411yescharacteristic, complemented
30C2×#103×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—
43#14519831—42no—
44(C2×#6923)⋊C4D412yes—
45C22×Dic122C2×C411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C2×#11758202—21no—
48C2×#11758202—21no—
49C2×#11758202—42no—
50C22×#92418C2×C411yescharacteristic
51C2×C8×D61—42no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56#14519757—21no—
57#14519757—21no—
58#14519757—21no—
59#14519757—21no—
60#14519753—21no—
61#14519753—21no—
62#14519753—21no—
63#14519753—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69#14519710—21no—
70#14519710—21no—
71#14519702—21no—
72#14519702—21no—
73C22×Q8×C61C2×C411yescharacteristic
74C22×C8×C61C2×C411yescharacteristic
75C2×C4⋊C4×C61C2×C411yescharacteristic
76#103×C61—21no—
77#103×C61—21no—
78C23×D61#4111yescharacteristic
79C22×#6923—22no—
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61C2×D411yescharacteristic
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×D411yescharacteristic
94C2×C4×D61C2×D411yescharacteristic
95C2×C4×D61#4111yescharacteristic
96C2×Dic122—21no—
97C2×Dic122—21no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100C22⋊#6923—42no—
101C2×#92420—22no—
102C2×#92420—41no—
103C2×#92420—41no—
104C4×#6923—41no—
105C4×#6923—41no—
106#11758213—41no—
107#11758213—41no—
108#11758213—41no—
109#11758213—41no—
110D61⋊C8—41no—
111D61⋊C8—41no—
112C2×C8×C61—41no—
113C2×C8×C61C22⋊C411yescharacteristic
114C2×C8×C61C22⋊C411yescharacteristic
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117#11758202—42no—
118#11758202—42no—
119#11758198—41no—
120#11758198—41no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127#11758196—41no—
128#11758196—41no—
129C2×#92418—22no—
130C2×#92418C22⋊C411yescharacteristic
131C2×#92418C22⋊C411yescharacteristic
132C8×D61—22no—
133C8×D61—22no—
134C8×D61—41no—
135C8×D61—41no—
136C23×C61C4×D411yescharacteristic
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—41no—
140C22×D61#12111yescharacteristic
141C22×D61#12411yescharacteristic
142C2×#6923—42no—
143C2×#6923—42no—
144C2×#6923—42no—
145Q8×C61—41no—
146Q8×C61—41no—
147Q8×C61—81no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—81no—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—81no—
161C2×Dic61#12111yescharacteristic
162C2×Dic61#12711yescharacteristic
163C4×D61—21no—
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—21no—
167C4×D61—41no—
168C4×D61—41no—
169Dic122—41no—
170Dic122—41no—
171Dic122—81no—
172#92420—42no—
173#92420—81no—
174#92420—81no—
175C8×C61—21no—
176C8×C61—21no—
177#92418—21no—
178#92418#9912yes—
179#9817—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61#39011yescharacteristic
183C22×C61#44412yes—
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—44no—
190C2×D61#47312yes—
191#6923—82no—
192C4×C61—21no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61C2×#9911yescharacteristic
196C4×C61#41011yescharacteristic
197Dic61—21no—
198Dic61—21no—
199Dic61—81no—
200Dic61—81no—
201C2×#467—611no—
202#1810—611no—
203#1752—611no—
204#1508—611no—
205#1508—611no—
206#1472—611no—
207#1472—611no—
208C2×C61—22no—
209C2×C61#196411yescharacteristic
210C2×C61#197511yescharacteristic
211C2×C61#197911yescharacteristic
212D61—22no—
213D61—41no—
214C22×Q16—611no—
215C23×C8—611no—
216C2×C22⋊Q8—611no—
217C2×C22⋊Q8—611no—
218#467—1221no—
219#467—1221no—
220#467—1221no—
221#467—1221no—
222C2×#108—1221no—
223C2×#107—611no—
224C2×#103—611no—
225C2×#103#692311yescharacteristic, complemented
226#407—611no—
227#407—611no—
228C2×C22⋊C8—611no—
229C2×#97—1221no—
230C4×C22⋊C4—611no—
231#341—1221no—
232#336—1221no—
233#328—1222no—
234#328—1222no—
235#324—1222no—
236#324—1222no—
237C61#981711yescharacteristic, complemented, Sylow
238C22×Q8—611no—
239C22×Q8C2×#692311yescharacteristic
240C23×C4—611no—
241C2×Q16—1221no—
242C2×Q16—1221no—
243C2×Q16—2442no—
244C2×M16—1221no—
245C2×M16—1221no—
246C22×C8—611no—
247C22×C8—611no—
248C22×C8—611no—
249C22×C8—611no—
250C22×C8—611no—
251C22×C8—1221no—
252C22×C8—2442no—
253C22×C8C2×#692311yescharacteristic
254C22⋊Q8—1221no—
255C22⋊Q8—1221no—
256C22⋊Q8—1221no—
257C22⋊Q8—1221no—
258C22⋊Q8—1221no—
259C22⋊Q8—1221no—
260C22⋊Q8—1221no—
261C22⋊Q8—1221no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C2×#692311yescharacteristic
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—1221no—
270#108—2442no—
271#107—1221no—
272#107—1221no—
273#103—21no—
274#103—21no—
275#103—1221no—
276#103—1221no—
277C22⋊C8—612no—
278C22⋊C8—1221no—
279C22⋊C8—1221no—
280C22⋊C8—1221no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283#97—2442no—
284#95—612no—
285C24—611no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×Q8—2441no—
293C2×Q8—2441no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—611no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—1221no—
302C22×C4—1222no—
303C22×C4—1221no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4C22×#692311yescharacteristic
307Q16—2442no—
308Q16—2442no—
309M16—2441no—
310M16—2441no—
311M16—2441no—
312M16—2441no—
313C2×C8—41no—
314C2×C8—611no—
315C2×C8—611no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—2441no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C2×C8—2441no—
325C2×C8—2441no—
326C2×C8C22⋊#692311yescharacteristic
327C2×C8C22⋊#692311yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C22⋊C4—2442no—
341C42—2441no—
342C42—2441no—
343C23—611no—
344C23—611no—
345C23—1222no—
346C23—1222no—
347C23—2441no—
348C23D4×#692311yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—81no—
352Q8—4881no—
353Q8—2441no—
354Q8—2441no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—4881no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4C2×C22⋊#692311yescharacteristic
378C2×C4D4×#692311yescharacteristic
379C8—21no—
380C8—21no—
381C8—612no—
382C8—1221no—
383C8—4881no—
384C8—4881no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—612no—
389C22—1221no—
390C22—1221no—
391C22—1221no—
392C22—1221no—
393C22—1221no—
394C22—2444no—
395C22#1684071411yescharacteristic
396C22#1684080712yes—
397C4—21no—
398C4—81no—
399C4—81no—
400C4—1221no—
401C4—1221no—
402C4—4882no—
403C4—4881no—
404C4—4881no—
405C4D61⋊#9911yescharacteristic
406C4#1684074111yescharacteristic
407C2—22no—
408C2—1222no—
409C2—2441no—
410C2#2146196011yescharacteristic
411C2#2146197511yescharacteristic
412C2#2146198111yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#103×C61C4
socleC22×C61#390
prime radicalC23×D61#41
cosocleC22×C4×D61C23
hypercenterC2×#103#6923
hypocenterC61#9817
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
20. C22×#11758202C4semidirect
28. C2×#14519710C4semidirect
29. C2×#14519702C4semidirect
30. C2×#103×C61C4semidirect
225. C2×#103#6923semidirect
237. C61#9817semidirect

Maximal subgroups

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

subgroupindex
#214616992
#214615552
#214612192
#214612182
#214610992
#214610972
C2×#168400022
#981761