Finite Group Databasebeta

← #30598402

Subgroups — #30598402

Order 15616.

738 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461685C211yesmaximal, characteristic, complemented
2#21461332C212yesmaximal, complemented
3#21461144C212yesmaximal
4#21461061C211yesmaximal, characteristic, complemented
5C2×#16840481C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519838—21no—
9#16840642—21no—
10#16840622—24no—
11D4⋊#92420—24no—
12#16840608—21no—
13#16840607—21no—
14#16840481—24no—
15#16840481—24no—
16C2×#14519775C2211yescharacteristic
17C2×#14519770—21no—
18C2×#14519770—21no—
19C2×#14519764C411yescharacteristic
20C2×#14519754—21no—
21C2×Q8×Dic61C411yescharacteristic
22C2×C4×Dic122C411yescharacteristic
23C2×C4×Dic122C2211yescharacteristic
24C2×D4×Dic61C2211yescharacteristic
25C2×#14519733—21no—
26C2×#124×C61C411yescharacteristic
27C2×#11758217—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C22×Dic122C2×C411yescharacteristic
35C22×Dic122D411yescharacteristic
36C23×Dic61—21no—
37#14519775—22no—
38#14519775D414yes—
39#14519770—28no—
40#14519770—28no—
41#14519764—22no—
42#14519764—22no—
43#14519754—28no—
44Q8×Dic61—22no—
45Q8×Dic61—22no—
46C4×Dic122—22no—
47C4×Dic122—22no—
48C4×Dic122—22no—
49C4×Dic122—22no—
50D4×Dic61—22no—
51D4×Dic61D414yes—
52#14519733—28no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196D411yescharacteristic
60C2×#11758196D411yescharacteristic
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C22×D4×C61C2×C411yescharacteristic
70#124×C61—22no—
71#124×C61—22no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C42×C61C2×C411yescharacteristic
75C24×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—42no—
78C2×Q8×C61C22⋊C412yes—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61C22⋊C412yes—
82C22×C4×C61—21no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—24no—
89C22×Dic61—42no—
90C22×Dic61—41no—
91C22×Dic61—41no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C2×Dic122—21no—
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—42no—
102C2×Dic122C22⋊C412yes—
103#11758217—42no—
104#11758217—42no—
105C2×#92420—42no—
106C2×#92420—42no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C42×C61—21no—
110C42×C61—21no—
111C22⋊Dic61—24no—
112C22⋊Dic61—24no—
113C22⋊Dic61—24no—
114C4⋊Dic61—24no—
115C4⋊Dic61—21no—
116C4⋊Dic61—22no—
117C4⋊Dic61C22⋊C412yes—
118#11758196—24no—
119#11758196—24no—
120#11758196—24no—
121#11758196—22no—
122#11758196—22no—
123C4×Dic61—24no—
124C4×Dic61—24no—
125C4×Dic61—21no—
126C4×Dic61—21no—
127C4×Dic61C2×D412yes—
128C4×Dic61C2×D412yes—
129C23×C61—22no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61C2×C22⋊C411yescharacteristic
135Q8×C61—42no—
136Q8×C61—42no—
137D4×C61—24no—
138D4×C61—42no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×Dic61—21no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—44no—
160C2×Dic61—44no—
161C2×Dic61—41no—
162C2×Dic61#12012yes—
163C2×Dic61#12012yes—
164C2×Dic61#13512yes—
165C2×Dic61AGL(1,ℤ/8)11yescharacteristic
166C2×Dic61#13711yescharacteristic
167Dic122—42no—
168Dic122—42no—
169Dic122—42no—
170Dic122—42no—
171#92420—82no—
172#92420—82no—
173#8394—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61#37911yescharacteristic
182C22×C61#41611yescharacteristic
183C22×C61#41711yescharacteristic
184C4×C61—21no—
185C4×C61—41no—
186C4×C61—42no—
187C4×C61—41no—
188C4×C61#41312yes—
189Dic61—22no—
190Dic61—41no—
191Dic61—42no—
192Dic61—41no—
193Dic61—42no—
194Dic61#44912yes—
195Dic61#45212yes—
196C2×#556—611no—
197#1801—611no—
198#1548—612no—
199#1486—612no—
200#1461—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—82no—
204C2×C61#178511yescharacteristic
205C2×C61#178811yescharacteristic
206C2×C61#179611yescharacteristic
207#556—1224no—
208#556—1224no—
209C2×C4⋊Q8—611no—
210C2×#126—1221no—
211C2×#126—1221no—
212C2×#124—611no—
213C2×#124#692311yescharacteristic
214C2×C22⋊Q8—1221no—
215C2×C4×Q8—611no—
216C2×C4×Q8—611no—
217C2×C4×Q8—611no—
218C2×C4×D4—611no—
219C2×#117—1221no—
220#431—612no—
221#423—612no—
222C2×#101—1221no—
223#343—1221no—
224#333—1224no—
225D4⋊C8—1224no—
226#324—1221no—
227#324—1221no—
228C61#839411yescharacteristic, complemented, Sylow
229C22×Q8—611no—
230C22×Q8—611no—
231C22×Q8C2×#692311yescharacteristic
232C22×D4C2×#692311yescharacteristic
233C23×C4—1221no—
234C2×M16—1222no—
235C4⋊Q8—614no—
236C4⋊Q8—1222no—
237#126—1228no—
238#126—1228no—
239#124—22no—
240#124—22no—
241#124—1222no—
242#124—1222no—
243C22⋊Q8—1228no—
244C4×Q8—1222no—
245C4×Q8—1222no—
246C4×Q8—1222no—
247C4×Q8—1222no—
248C4×Q8—1222no—
249C4×Q8—1222no—
250C4×D4—614no—
251C4×D4—1222no—
252#117—1228no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C42—611no—
267C2×C42—611no—
268C2×C42—1221no—
269C2×C42—1221no—
270C2×C42C2×#692311yescharacteristic
271C4⋊C8—1222no—
272C4⋊C8—1222no—
273C4⋊C8—1222no—
274#101—2442no—
275C22⋊C8—1222no—
276C4×C8—1222no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—612no—
281C2×Q8—1221no—
282C2×Q8—1222no—
283C2×Q8—1222no—
284C2×Q8—1221no—
285C2×Q8—1221no—
286C2×Q8—2442no—
287C2×Q8C22⋊#692312yes—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4C22⋊#692312yes—
291C22×C4—21no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—1224no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—2442no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305M16—2442no—
306M16—2442no—
307C2×C8—2442no—
308C2×C8—2442no—
309C4⋊C4—612no—
310C4⋊C4—1222no—
311C4⋊C4—1222no—
312C4⋊C4—1222no—
313C4⋊C4—1224no—
314C4⋊C4—1224no—
315C4⋊C4—1224no—
316C4⋊C4—1224no—
317C4⋊C4—1221no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—1224no—
321C22⋊C4—1224no—
322C22⋊C4—1224no—
323C42—21no—
324C42—21no—
325C42—612no—
326C42—612no—
327C42—1224no—
328C42—1224no—
329C42—1221no—
330C42—1221no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23C2×C22⋊#692311yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—2442no—
340Q8—2442no—
341Q8—2442no—
342Q8—2442no—
343D4—24no—
344D4—42no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—612no—
354C2×C4—611no—
355C2×C4—611no—
356C2×C4—612no—
357C2×C4—612no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1222no—
362C2×C4—1221no—
363C2×C4—1222no—
364C2×C4—2444no—
365C2×C4—2444no—
366C2×C4—2441no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4C2×C22⋊#692311yescharacteristic
372C2×C4C2×C22⋊#692311yescharacteristic
373C8—4882no—
374C8—4882no—
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22#1684070311yescharacteristic
383C22#1684075211yescharacteristic
384C22#1684075311yescharacteristic
385C4—21no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—612no—
390C4—612no—
391C4—1222no—
392C4—2442no—
393C4—2441no—
394C4—2442no—
395C4—2441no—
396C4#1684074912yes—
397C2—21no—
398C2—21no—
399C2—82no—
400C2#2146165611yescharacteristic
401C2#2146165911yescharacteristic
402C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC22×C61#379
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#8394
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461685C2semidirect
2. #21461332C2semidirect
4. #21461061C2semidirect
228. C61#8394semidirect

Maximal subgroups

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

subgroupindex
#214616852
#214613322
#214611442
#214610612
C2×#168404812
#839461