Finite Group Databasebeta

← #30605475

Subgroups — #30605475

Order 15616.

508 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461681C211yesmaximal, characteristic
2#21461057C211yesmaximal, characteristic
3#21460816C212yesmaximal, complemented
4#21460815C212yesmaximal, complemented
5#21450767C211yesmaximal, characteristic
6C2×#14519835—21no—
7C2×C22⋊#92420—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519776C411yescharacteristic, complemented
13C2×#14519755—21no—
14C2×#14519754—21no—
15C2×#14519737C2211yescharacteristic
16#16839699C2211yescharacteristic
17#16839698C411yescharacteristic, complemented
18#16839697C2211yescharacteristic
19#16839662—21no—
20#16839631—21no—
21#16839613C411yescharacteristic, complemented
22C61×#399C411yescharacteristic, complemented
23C22×#92420—41no—
24C2×C4⋊#6923—41no—
25C2×C22⋊#6923—22no—
26#14519835—42no—
27C22⋊#92420—22no—
28C22⋊#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×Dic122—21no—
31C22×C22⋊D61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34#14519776—41no—
35#14519776—41no—
36#14519755—44no—
37#14519754—44no—
38#14519737—41no—
39#14519737—41no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196D411yescharacteristic
47C2×C22⋊Dic61—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198C2311yescharacteristic
51C2×C4×Dic61—21no—
52C2×C4×Dic61C2×C411yescharacteristic
53#14519684C2×C411yescharacteristic
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61C2×C411yescharacteristic
58#95×C61C2×C411yescharacteristic
59C24×C61C22⋊C411yescharacteristic
60C23×D61C2×D411yescharacteristic
61C22×#6923—42no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C2×C22⋊D61—41no—
67C2×C22⋊D61—41no—
68C2×C22⋊D61SD1612yes—
69C2×C22⋊D61Q1612yes—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—42no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—42no—
81C2×Dic122—42no—
82C2×Dic122—42no—
83C2×Dic122—41no—
84C2×Dic122—41no—
85C22⋊#6923—42no—
86C22⋊#6923—42no—
87C2×#92420—42no—
88C2×#92420—82no—
89C4⋊#6923—82no—
90C4⋊C4×C61—42no—
91C4⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C22⋊Dic61—42no—
95#11758198—41no—
96#11758198—42no—
97#11758198—42no—
98C4⋊Dic61—42no—
99C4⋊Dic61—41no—
100#11758196—24no—
101#11758196—41no—
102#11758196—42no—
103#11758196—42no—
104#11758196—42no—
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61—42no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10312yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61C2×SD1611yescharacteristic
119C22×D61C2×Q1611yescharacteristic
120C2×#6923—82no—
121C2×#6923—82no—
122C2×#6923—82no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C22⋊D61—42no—
134C22⋊D61—42no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—22no—
138C2×Dic61—22no—
139C2×Dic61—21no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—81no—
148C2×Dic61#9911yescharacteristic
149C2×Dic61#9911yescharacteristic
150C2×Dic61#10312yes—
151C4×D61—42no—
152C4×D61—81no—
153Dic122—82no—
154Dic122—82no—
155#92420—82no—
156#11901—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—22no—
160C22×C61—42no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61—81no—
164C22×C61C2×#9911yescharacteristic
165C22×C61C2×#10311yescharacteristic
166C22×C61#42111yescharacteristic
167C2×D61—21no—
168C2×D61—21no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61—81no—
174#6923—162no—
175C4×C61—42no—
176C4×C61—41no—
177C4×C61—41no—
178C4×C61—81no—
179Dic61—21no—
180Dic61—22no—
181Dic61—42no—
182Dic61—41no—
183Dic61—81no—
184Dic61#35312yes—
185#2542—611no—
186#1799—611no—
187#1459—611no—
188#1352—612no—
189#1352—612no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61#152011yescharacteristic
194C2×C61#212311yescharacteristic
195C2×C61C2×#35311yescharacteristic
196D61—41no—
197D61—41no—
198C2×C4⋊Q8—611no—
199C2×C22⋊Q8—1221no—
200C2×C22⋊Q8—1221no—
201C2×C4×D4—611no—
202C2×#99—1221no—
203C2×C22⋊C8—1221no—
204#399—611no—
205#399—611no—
206#399—611no—
207#399#692311yescharacteristic, complemented
208#390—1221no—
209#384—1221no—
210#380—1221no—
211C4×C4⋊C4—611no—
212#342—612no—
213#324—1221no—
214#323—612no—
215C61#1190111yescharacteristic, complemented, Sylow
216C22×Q8—1221no—
217C22×D4—611no—
218C23×C4—611no—
219C23×C4—611no—
220C22×C8—2441no—
221C4⋊Q8—2441no—
222C4⋊Q8—2441no—
223C22⋊Q8—2444no—
224C22⋊Q8—2444no—
225C4×D4—2441no—
226C4×D4—2441no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—2441no—
236C2×C4⋊C4C2×#692311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—611no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4C2×#692311yescharacteristic
244C2×C42—611no—
245C2×C42—1221no—
246#99—2442no—
247C22⋊C8—1222no—
248C22⋊C8—2441no—
249#95—611no—
250#95—2441no—
251#95C2×#692311yescharacteristic
252C24—611no—
253C24C22⋊#692311yescharacteristic
254C2×Q8—2441no—
255C2×Q8—2441no—
256C2×Q8—2442no—
257C2×Q8—2442no—
258C2×D4—612no—
259C2×D4—612no—
260C2×D4—2441no—
261C2×D4—2441no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—2442no—
274C22×C4—2442no—
275C22×C4—2442no—
276C22×C4C22⋊#692311yescharacteristic
277C22×C4C22×#692311yescharacteristic
278C2×C8—4882no—
279C2×C8—2442no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—1224no—
283C4⋊C4—4882no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C4⋊C4—2442no—
287C4⋊C4—2442no—
288C4⋊C4—2442no—
289C4⋊C4—2442no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—2441no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C22⋊C4—2442no—
296C22⋊C4—2442no—
297C22⋊C4—2442no—
298C42—1221no—
299C42—1221no—
300C42—2442no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—611no—
306C23—611no—
307C23—1221no—
308C23—1221no—
309C23—2441no—
310C23—2441no—
311C23#1451984012yes—
312C23C2×C22⋊#692311yescharacteristic
313Q8—4882no—
314Q8—4882no—
315D4—2442no—
316D4—2442no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—612no—
330C2×C4—1222no—
331C2×C4—1221no—
332C2×C4—1221no—
333C2×C4—1222no—
334C2×C4—1221no—
335C2×C4—4881no—
336C2×C4—4881no—
337C2×C4—4882no—
338C2×C4—4882no—
339C2×C4—4882no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C8—4882no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—81no—
356C22—1221no—
357C22—1221no—
358C22—4881no—
359C22—2441no—
360C22—2441no—
361C22—2441no—
362C22—2441no—
363C22D61⋊#9911yescharacteristic
364C22C2×#1451984011yescharacteristic
365C22#1684075911yescharacteristic
366C4—42no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—612no—
371C4—1222no—
372C4—1221no—
373C4—9762no—
374C4—4881no—
375C4—2442no—
376C4—2441no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—2441no—
381C2—2441no—
382C2#2146125711yescharacteristic
383C2#2146208211yescharacteristic
384C2D61⋊#35311yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenter#399#6923
hypocenterC61#11901
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
4. #21460815C2semidirect
12. C2×#14519776C4semidirect
17. #16839698C4semidirect
21. #16839613C4semidirect
22. C61×#399C4semidirect
207. #399#6923semidirect
215. C61#11901semidirect

Maximal subgroups

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

subgroupindex
#214616812
#214610572
#214608162
#214608152
#214507672
#1190161