Finite Group Databasebeta

← #30606819

Subgroups — #30606819

Order 15616.

1098 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 1098 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
1C2×#16840668C212yesmaximal, complemented
2C2×#16840666C212yesmaximal, complemented
3#21461648C212yesmaximal, complemented
4C2×#16840308C211yesmaximal, characteristic
5C2×#14519836C2214yes—
6C2×#14519835—22no—
7#16840705—22no—
8#16840668—24no—
9#16840666—24no—
10C22×#11758222C2211yescharacteristic
11#16840308—24no—
12#16840308—24no—
13C2×#14519775C411yescharacteristic, complemented
14C2×#14519773C2211yescharacteristic
15C2×#14519766C2211yescharacteristic
16C2×#14519764C411yescharacteristic, complemented
17C2×#14519749—22no—
18C2×C4×C22⋊D61—22no—
19C22×C4×Dic61C411yescharacteristic, complemented
20C22×#41×C61C411yescharacteristic, complemented
21C2×C22⋊#6923—24no—
22#14519836—24no—
23#14519836—24no—
24#14519835—44no—
25(C2×#6923)⋊C4—44no—
26C2×#11758222—22no—
27C2×#11758222—22no—
28C2×#11758222—42no—
29C2×#11758222—42no—
30C22×Dic122C2×C411yescharacteristic
31C22×D244C2311yescharacteristic
32C22×C22⋊D61—22no—
33C22×C22⋊D61D412yes—
34C23×Dic61—21no—
35C22×C4×D61D412yes—
36#14519775—22no—
37#14519775—22no—
38#14519773—22no—
39#14519773D414yes—
40#14519766—22no—
41#14519766D414yes—
42#14519764—22no—
43#14519764—22no—
44#14519749—216no—
45C4×C22⋊D61—216no—
46C2×#11758196—22no—
47C2×C22⋊Dic61—22no—
48C2×#11758198—22no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—22no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61C2×C411yescharacteristic
54C2×#41×C61—22no—
55C2×#41×C61—22no—
56C2×#41×C61—42no—
57C2×#41×C61—42no—
58C22×Q8×C61C2×C411yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61C2×C411yescharacteristic
61C23×C4×C61—21no—
62C23×C4×C61C2×C411yescharacteristic
63C24×C61—21no—
64C24×C61C22⋊C411yescharacteristic
65C23×D61C2×D412yes—
66C22×#6923—44no—
67#41×C61—48no—
68#41×C61—48no—
69C2×Q8×C61—21no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61C22⋊C412yes—
73C2×D4×C61—24no—
74C2×D4×C61—21no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61—42no—
78C2×D4×C61—42no—
79C2×D4×C61C22⋊C412yes—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—24no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22×C411yescharacteristic
90C2×C22⋊D61—24no—
91C2×C22⋊D61—28no—
92C2×C22⋊D61—24no—
93C2×C22⋊D61—42no—
94C2×C22⋊D61—48no—
95C22×Dic61—21no—
96C22×Dic61—24no—
97C22×Dic61—21no—
98C22×Dic61—44no—
99C22×Dic61C22⋊C412yes—
100#11758222—48no—
101#11758222—48no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244C2×D412yes—
105C2×D244C2×D412yes—
106C2×C4×D61—24no—
107C2×C4×D61—24no—
108C2×C4×D61—42no—
109C2×Dic122—21no—
110C2×Dic122—21no—
111C2×Dic122—41no—
112C2×Dic122—41no—
113C22⋊#6923—44no—
114C22⋊#6923—44no—
115C22⋊Dic61—28no—
116#11758198—28no—
117#11758196—28no—
118C4×Dic61—24no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61C22⋊C412yes—
122C4×Dic61C22⋊C412yes—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—22no—
126C23×C61—42no—
127C23×C61—44no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61#9912yes—
131C23×C61C2×C22⋊C411yescharacteristic
132C22×D61—22no—
133C22×D61—22no—
134C22×D61—44no—
135C22×D61#12014yes—
136C2×#6923—84no—
137C2×#6923—88no—
138Q8×C61—42no—
139Q8×C61—42no—
140D4×C61—42no—
141D4×C61—42no—
142D4×C61—48no—
143C2×C4×C61—24no—
144C2×C4×C61—24no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—48no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61#9912yes—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C2×C22⋊C411yescharacteristic
156C22⋊D61—416no—
157C22⋊D61—48no—
158C2×Dic61—24no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—48no—
165C2×Dic61—42no—
166D244—42no—
167D244—42no—
168C4×D61—48no—
169Dic122—42no—
170Dic122—42no—
171#12695—611nomaximal, Sylow
172C22×C61—22no—
173C22×C61—22no—
174C22×C61—42no—
175C22×C61—42no—
176C22×C61—44no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61#37911yescharacteristic
182C22×C61C2×#9911yescharacteristic
183C22×C61C2×#9911yescharacteristic
184C2×D61—24no—
185C2×D61—44no—
186C2×D61—44no—
187C2×D61—82no—
188#6923—164no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—22no—
192C4×C61—42no—
193Dic61—24no—
194Dic61—42no—
195Dic61—42no—
196C2×#532—611no—
197C2×#354—612no—
198C2×#353—612no—
199#1782—612no—
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—42no—
203C2×C61—82no—
204C2×C61#178111yescharacteristic
205C2×C61#212612yes—
206D61—44no—
207C22×#41—611no—
208C22×#41#692311yescharacteristic, complemented
209#532—1224no—
210#532—1224no—
211C2×C4⋊Q8—611no—
212C2×#127—611no—
213C2×#124—611no—
214C2×#124—611no—
215C2×#121—1222no—
216C2×C4×D4—1222no—
217C22×C42—611no—
218C2×#99—614no—
219C2×#99—1222no—
220#381—1222no—
221#354—1224no—
222#353—1224no—
223C61#1269511yescharacteristic, complemented, Sylow
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—1222no—
229C2×#41—1222no—
230C2×#41—2442no—
231C2×#41—2442no—
232C22×Q8—611no—
233C22×Q8C2×#692311yescharacteristic
234C22×D4—21no—
235C22×D4—612no—
236C22×D4—611no—
237C22×D4—1222no—
238C22×D4C2×#692311yescharacteristic
239C23×C4—21no—
240C23×C4—612no—
241C23×C4—1221no—
242C23×C4C2×#692311yescharacteristic
243C4⋊Q8—1222no—
244C4⋊Q8—1222no—
245#127—614no—
246#127—1222no—
247#124—614no—
248#124—1222no—
249#124—1222no—
250#124—1222no—
251#121—12216no—
252C4×D4—12216no—
253C2×C4⋊C4—1222no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1222no—
256C2×C22⋊C4—1224no—
257C2×C42—611no—
258C2×C42—611no—
259C2×C42—1222no—
260C2×C42—1222no—
261C2×C42—1221no—
262#99—1224no—
263#99—1224no—
264#99—2444no—
265#95—2444no—
266C24—21no—
267C24—612no—
268C24C22⋊#692311yescharacteristic
269#41—48no—
270#41—48no—
271#41—2448no—
272#41—2448no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—1221no—
277C2×Q8—1221no—
278C2×Q8—2441no—
279C2×Q8—2441no—
280C2×Q8C22⋊#692312yes—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—612no—
288C2×D4—612no—
289C2×D4—1228no—
290C2×D4—1224no—
291C2×D4—1224no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2442no—
295C2×D4—2448no—
296C2×D4C22⋊#692312yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—612no—
306C22×C4—1224no—
307C22×C4—1221no—
308C22×C4—1224no—
309C22×C4—1221no—
310C22×C4—1224no—
311C22×C4—2442no—
312C22×C4—2444no—
313C22×C4—2444no—
314C22×C4C22⋊#692311yescharacteristic
315C22×C4C22×#692311yescharacteristic
316C4⋊C4—1228no—
317C22⋊C4—1228no—
318C22⋊C4—1228no—
319C22⋊C4—2444no—
320C22⋊C4—2444no—
321C42—612no—
322C42—612no—
323C42—1224no—
324C42—1221no—
325C42—1221no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—42no—
330C23—44no—
331C23—41no—
332C23—41no—
333C23—614no—
334C23—1222no—
335C23—1222no—
336C23—2444no—
337C23#1451983512yes—
338C23C2×C22⋊#692311yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—2442no—
342Q8—2442no—
343D4—42no—
344D4—42no—
345D4—48no—
346D4—2442no—
347D4—2442no—
348D4—24416no—
349D4—2448no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—48no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—1224no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—1222no—
366C2×C4—4884no—
367C2×C4—4888no—
368C2×C4—2448no—
369C2×C4—2448no—
370C2×C4—2442no—
371C2×C4#1451983512yes—
372C2×C4C2×C22⋊#692311yescharacteristic
373C2×C4C2×C22⋊#692311yescharacteristic
374C22—22no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22—1224no—
384C22—4882no—
385C22—2444no—
386C22—2444no—
387C22#1684070311yescharacteristic
388C22C2×#1451983511yescharacteristic
389C22C2×#1451983511yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—1224no—
395C4—9764no—
396C4—2442no—
397C4—2442no—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—82no—
402C2—2444no—
403C2#2146164511yescharacteristic
404C2#2146209012yes—
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×#41×C61C4
socleC22×C61C2×#99
prime radicalC2×#16840308C2
cosocleC22×D244C23
hypercenterC22×#41#6923
hypocenterC61#12695
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840668C2semidirect
2. C2×#16840666C2semidirect
3. #21461648C2semidirect
13. C2×#14519775C4semidirect
16. C2×#14519764C4semidirect
19. C22×C4×Dic61C4semidirect
20. C22×#41×C61C4semidirect
208. C22×#41#6923semidirect
223. C61#12695semidirect

Maximal subgroups

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

subgroupindex
C2×#168406682
C2×#168406662
#214616482
C2×#168403082
#1269561