Finite Group Databasebeta

← #30600623

Subgroups — #30600623

Order 15616.

474 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 90 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#21462278C211yesmaximal, characteristic
2#21462004C211yesmaximal, characteristic
3#21461203C211yesmaximal, characteristic
4#21461148C211yesmaximal, characteristic
5#21460716C211yesmaximal, characteristic
6#21460646C211yesmaximal, characteristic
7#21455860C211yesmaximal, characteristic
8#16840781C2211yescharacteristic
9#16840766C2211yescharacteristic
10(C2×#6923)⋊Q8—21no—
11#16840713—21no—
12#16840711—22no—
13C4×C4⋊#6923—21no—
14C4×C4⋊#6923C2211yescharacteristic
15#16840698—21no—
16C42×#6923C2211yescharacteristic
17#16840632—21no—
18#16840632—21no—
19#16840616—22no—
20#16840612—21no—
21Q8⋊#92420—21no—
22#16840317C2211yescharacteristic
23C4×Q8×D61C2211yescharacteristic
24#16840099—21no—
25#16840054C411yescharacteristic, complemented
26#16839973C411yescharacteristic, complemented
27#16839972C411yescharacteristic, complemented
28#16839919—21no—
29#16839856—21no—
30#16839854—21no—
31#16839814C2211yescharacteristic
32C61×#477C411yescharacteristic, complemented
33C2×#11758213—21no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—21no—
36C2×C4×#6923—21no—
37C2×C4×#6923—21no—
38C2×C4×#6923—21no—
39C2×C4×#6923D411yescharacteristic
40C2×C4×#6923D411yescharacteristic
41C4⋊#92420—21no—
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44#14519834—21no—
45C4×#92420—21no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—22no—
49(C2×#6923)⋊C4D412yes—
50C2×Q8×D61D411yescharacteristic
51C2×Dic244—21no—
52C2×#11758202—21no—
53#14519777C2×C411yescharacteristic
54#14519768—21no—
55#14519757—21no—
56#14519753—21no—
57Q8×Dic61—21no—
58Q8×Dic61D411yescharacteristic
59C4×Dic122—21no—
60C4×Dic122C2×C411yescharacteristic
61#14519742—21no—
62#14519742—21no—
63#14519742D411yescharacteristic
64#14519735—21no—
65C4⋊C4×D61—21no—
66C4⋊C4×D61D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519710—21no—
70C4⋊#92418C2×C411yescharacteristic
71#14519702—21no—
72#14519700—21no—
73#14519700—21no—
74#14519689—21no—
75C8×Dic61—21no—
76C4⋊Q8×C61C2×C411yescharacteristic
77C4×Q8×C61C2×C411yescharacteristic
78#107×C61—21no—
79C4⋊C8×C61C2×C411yescharacteristic
80#103×C61—21no—
81C22×#6923—21no—
82C22×#6923—21no—
83C22×#6923—22no—
84C2×Q8×C61—21no—
85C2×Q8×C61C22⋊C411yescharacteristic
86Q8×D61—41no—
87Q8×D61—41no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61C2×D411yescharacteristic
91C2×C4×D61C2×D411yescharacteristic
92C2×C4×D61C2×D411yescharacteristic
93C2×Dic122—21no—
94C2×Dic122—21no—
95C2×Dic122C22⋊C411yescharacteristic
96C2×#92420—41no—
97C2×#92420—41no—
98C4⋊#6923—41no—
99C4⋊#6923—41no—
100C4⋊#6923—44no—
101C4×#6923—21no—
102C4×#6923—21no—
103C4×#6923—21no—
104C4×#6923—41no—
105C4×#6923—41no—
106C4×#6923—42no—
107C4×#6923—41no—
108C4×#6923—41no—
109C4×#6923Q1612yescomplemented
110#11758213—41no—
111#11758213—41no—
112C2×C8×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61C22⋊C411yescharacteristic
117C42×C61—21no—
118C42×C61C22×C411yescharacteristic
119#11758202—42no—
120#11758198—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—41no—
125C4⋊Dic61C22⋊C411yescharacteristic
126#11758196—21no—
127#11758196—21no—
128#11758196—21no—
129#11758196—41no—
130C4×Dic61—21no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133C4×Dic61—41no—
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61#4111yescharacteristic
136C4×Dic61#4111yescharacteristic
137C2×#92418—21no—
138Dic244—42no—
139C22×D61#12011yescharacteristic
140C2×#6923—22no—
141C2×#6923—22no—
142C2×#6923—41no—
143C2×#6923—41no—
144C2×#6923—41no—
145C2×#6923—41no—
146C2×#6923—42no—
147C2×#6923—42no—
148C2×#6923—42no—
149Q8×C61—21no—
150Q8×C61—21no—
151Q8×C61—81no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61C2×C22⋊C411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—41no—
161C2×Dic61#12111yescharacteristic
162C2×Dic61#12111yescharacteristic
163C2×Dic61#12311yescharacteristic
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—41no—
167C4×D61—42no—
168C4×D61—42no—
169C4×D61C2×Q1611yescharacteristic
170C4×D61#13511yescharacteristic
171C4×D61#13711yescharacteristic
172C4×D61#13711yescharacteristic
173Dic122—21no—
174Dic122—21no—
175Dic122—42no—
176Dic122—41no—
177Dic122—41no—
178#92420—81no—
179#92420—81no—
180C8×C61—41no—
181#92418—41no—
182#9236—611nomaximal, Sylow
183C22×C61#38611yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61#45111yescharacteristic
187C2×D61#45211yescharacteristic
188#6923—42no—
189#6923—42no—
190#6923—82no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—81no—
195C4×C61#42111yescharacteristic
196C4×C61#42111yescharacteristic
197C4×C61C4×Q1611yescharacteristic
198C4×C61#44111yescharacteristic
199Dic61—21no—
200Dic61—21no—
201Dic61—41no—
202Dic61—41no—
203Dic61—81no—
204Dic61#46211yescharacteristic
205Dic61#46411yescharacteristic
206#3104—611no—
207#2295—611no—
208#1994—611no—
209#1503—611no—
210#1487—611no—
211#1303—611no—
212#1276—611no—
213C2×C61#187811yescharacteristic
214C2×C61#189711yescharacteristic
215C2×C61#190611yescharacteristic
216D61—21no—
217D61—21no—
218#533—611no—
219C2×C4×Q8—611no—
220#484—1221no—
221#477—611no—
222#477#692311yescharacteristic, complemented
223#462—611no—
224#462—611no—
225#451—1221no—
226C4×Q16—1221no—
227C4×Q16—1221no—
228#431—611no—
229#431—611no—
230#423—611no—
231#391—1221no—
232#389—1221no—
233#387—1222no—
234C4×C4⋊C4—611no—
235C4×C4⋊C4—1221no—
236#376—1221no—
237C43—611no—
238#339—1221no—
239#339—1221no—
240#328—1222no—
241Q8⋊C8—1221no—
242Q8⋊C8—1221no—
243C61#923611yescharacteristic, complemented, Sylow
244C22×Q8—611no—
245C2×Q16—1221no—
246C2×Q16—1221no—
247C2×M16—1221no—
248C4⋊Q8—611no—
249C4⋊Q8C2×#692311yescharacteristic
250#125—1221no—
251C22⋊Q8—1221no—
252C22⋊Q8—1221no—
253C4×Q8—611no—
254C4×Q8—611no—
255C4×Q8—611no—
256C4×Q8—1221no—
257C4×Q8—1221no—
258C4×Q8—1221no—
259C4×Q8—1221no—
260C4×Q8C2×#692311yescharacteristic
261#117—1221no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1222no—
266C2×C42—611no—
267C2×C42—611no—
268C2×C42—611no—
269C2×C42—1221no—
270C2×C42—1221no—
271C2×C42—1221no—
272C2×C42—1221no—
273#107—21no—
274#107—1221no—
275C4⋊C8—611no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C4⋊C8C2×#692311yescharacteristic
280#103—21no—
281#103—1221no—
282#103—1221no—
283#103—1221no—
284C22⋊C8—1221no—
285C22⋊C8—1221no—
286C4×C8—1221no—
287C4×C8—1221no—
288#95—612no—
289#95—1222no—
290#95—1222no—
291#95—1222no—
292C2×Q8—21no—
293C2×Q8—611no—
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—2441no—
297C2×Q8—2441no—
298C2×Q8C22⋊#692311yescharacteristic
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—611no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1222no—
307Q16—2442no—
308Q16—2442no—
309M16—2441no—
310M16—2441no—
311C2×C8—21no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8—2441no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—611no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C4⋊C4—2444no—
330C4⋊C4C22⋊#692311yescharacteristic
331C22⋊C4—2441no—
332C42—21no—
333C42—611no—
334C42—611no—
335C42—611no—
336C42—612no—
337C42—1221no—
338C42—1221no—
339C42—1221no—
340C42—1221no—
341C42—1221no—
342C42—1221no—
343C42—2441no—
344C42—2441no—
345C42—2441no—
346C42—2441no—
347C42—2441no—
348C42—2442no—
349C42C22×#692311yescharacteristic
350C23—611no—
351Q8—21no—
352Q8—21no—
353Q8—81no—
354Q8—1221no—
355Q8—1221no—
356Q8—2442no—
357Q8—2441no—
358Q8—2441no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—611no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4C2×C22⋊#692311yescharacteristic
387C2×C4D4×#692311yescharacteristic
388C2×C4D4×#692311yescharacteristic
389C8—41no—
390C8—4881no—
391C8—4881no—
392C8—2441no—
393C22—611no—
394C22—611no—
395C22—1221no—
396C22—1221no—
397C22#1684071011yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—611no—
403C4—611no—
404C4—1221no—
405C4—1221no—
406C4—4882no—
407C4—4881no—
408C4—2442no—
409C4—2442no—
410C4—2441no—
411C4—2441no—
412C4#1684075911yescharacteristic
413C4#1684076111yescharacteristic
414C4Q16×#692311yescharacteristic
415C4#1684080211yescharacteristic
416C2—1221no—
417C2—1221no—
418C2#2146181311yescharacteristic
419C2#2146183811yescharacteristic
420C2#2146185611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#477C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#477#6923
hypocenterC61#9236
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
25. #16840054C4semidirect
26. #16839973C4semidirect
27. #16839972C4semidirect
32. C61×#477C4semidirect
109. C4×#6923Q16semidirect
222. #477#6923semidirect
243. C61#9236semidirect

Maximal subgroups

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

subgroupindex
#214622782
#214620042
#214612032
#214611482
#214607162
#214606462
#214558602
#923661