Finite Group Databasebeta

← #30601804

Subgroups — #30601804

Order 15616.

590 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461735C211yesmaximal, characteristic, complemented
2#21461604C211yesmaximal, characteristic, complemented
3C2×#16840621C212yesmaximal, complemented
4C2×#16840620C212yesmaximal
5#21454173C211yesmaximal, characteristic
6C2×#14519844C2212yes—
7C2×#14519844C2212yes—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628—21no—
11#16840628—21no—
12#16840621—24no—
13#16840620—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519773C2211yescharacteristic
16C2×#14519763—21no—
17C2×#14519763—21no—
18#16839665C411yescharacteristic
19#16839664C411yescharacteristic
20#16839663C411yescharacteristic
21#16839662—21no—
22#16839662—21no—
23C42×Dic61C2211yescharacteristic
24C61×#390C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844D414yes—
31#14519832—42no—
32C22×Dic122—21no—
33C22×D244C2×C411yescharacteristic
34C22×C22⋊D61—21no—
35C22×C22⋊D61—21no—
36#14519776—22no—
37#14519776—22no—
38#14519773—22no—
39#14519773—22no—
40#14519763—44no—
41#14519763—44no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61C2311yescharacteristic
56C22×D4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C42×C61C2×C411yescharacteristic
61C24×C61—21no—
62C23×D61—21no—
63C2×D4×C61—22no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C2×C22⋊D61—42no—
71C2×C22⋊D61—41no—
72C2×C22⋊D61—41no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—41no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C2×C22⋊D61—41no—
78C22×Dic61—21no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×D244—22no—
84C2×D244—41no—
85C2×D244—41no—
86C2×Dic122—41no—
87C2×Dic122—41no—
88C2×Dic122—42no—
89C2×Dic122—42no—
90C2×#92420—24no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—22no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C42×C61—41no—
100C22⋊Dic61—42no—
101C22⋊Dic61—42no—
102#11758198—42no—
103#11758198—42no—
104#11758198—42no—
105#11758198—42no—
106C4⋊Dic61—22no—
107#11758196—42no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61—22no—
111C4×Dic61—22no—
112C4×Dic61—21no—
113C4×Dic61—41no—
114C4×Dic61C2×D412yes—
115C4×Dic61#4112yes—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131D4×C61—82no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61C4×D412yes—
144C22⋊D61—82no—
145C22⋊D61—82no—
146C22⋊D61—82no—
147C22⋊D61—82no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61#12112yes—
158C2×Dic61#12411yescharacteristic
159C2×Dic61#12711yescharacteristic
160C2×Dic61#13512yes—
161C2×Dic61AGL(1,ℤ/8)11yescharacteristic
162C2×Dic61AGL(1,ℤ/8)12yes—
163C2×Dic61AGL(1,ℤ/8)12yes—
164C2×Dic61#13711yescharacteristic
165D244—82no—
166Dic122—82no—
167Dic122—82no—
168#92420—44no—
169#92420—82no—
170#9710—611nomaximal, Sylow
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61#39011yescharacteristic
179C22×C61#41311yescharacteristic
180C22×C61#41311yescharacteristic
181C22×C61#41611yescharacteristic
182C22×C61#41711yescharacteristic
183C22×C61#41811yescharacteristic
184C22×C61#41811yescharacteristic
185C2×D61—81no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C4×C61—22no—
193C4×C61—41no—
194C4×C61—41no—
195C4×C61—81no—
196Dic61—41no—
197Dic61—41no—
198Dic61—81no—
199Dic61#46312yes—
200Dic61#46412yes—
201Dic61#49012yes—
202Dic61#49612yes—
203#2858—611no—
204#1831—611no—
205#1767—611no—
206C2×#332—612no—
207C2×#331—612no—
208C2×C61—161no—
209C2×C61#155311yescharacteristic
210C2×C61#155311yescharacteristic
211C2×C61#155711yescharacteristic
212C2×C61#155711yescharacteristic
213C2×C61#196111yescharacteristic
214C2×C61#196711yescharacteristic
215C2×C61#198311yescharacteristic
216D61—161no—
217C2×C4⋊Q8—611no—
218C2×#127—611no—
219C2×#124—1221no—
220C2×#124—1221no—
221C2×C4⋊C8—612no—
222C2×C4⋊C8—612no—
223C2×#97—1221no—
224#390—611no—
225#390—611no—
226#390—611no—
227#390—1221no—
228#390—1221no—
229#390#692311yescharacteristic
230C43—611no—
231#336—1221no—
232#336—1221no—
233#336—1221no—
234#332—1224no—
235#331—1224no—
236C61#971011yescharacteristic, complemented, Sylow
237C22×Q8—1221no—
238C22×D4—611no—
239C22×D4—1221no—
240C22×D4—1221no—
241C22×D4C2×#692311yescharacteristic
242C22×C8—1222no—
243C22×C8—1222no—
244C4⋊Q8—1222no—
245C4⋊Q8—1222no—
246#127—1222no—
247#127—1222no—
248#124—2444no—
249#124—2444no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C2×#692311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42—611no—
262C2×C42—611no—
263C2×C42—611no—
264C2×C42—611no—
265C2×C42—611no—
266C2×C42—611no—
267C2×C42C2×#692311yescharacteristic
268C4⋊C8—614no—
269C4⋊C8—1222no—
270C4⋊C8—1222no—
271C4⋊C8—1222no—
272#97—2442no—
273C24—21no—
274C24—1221no—
275C2×Q8—2442no—
276C2×Q8—2442no—
277C2×Q8—2441no—
278C2×Q8—2441no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—1222no—
283C2×D4—2441no—
284C2×D4—2441no—
285C2×D4—2442no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2441no—
289C2×D4—2441no—
290C2×D4—2442no—
291C2×D4—2441no—
292C2×D4—2441no—
293C22×C4—21no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—611no—
298C22×C4—1221no—
299C22×C4C22⋊#692311yescharacteristic
300C22×C4C22⋊#692311yescharacteristic
301C22×C4C22×#692311yescharacteristic
302C2×C8—1224no—
303C2×C8—2442no—
304C2×C8—2442no—
305C2×C8—2442no—
306C2×C8—2442no—
307C2×C8—2442no—
308C4⋊C4—22no—
309C4⋊C4—1222no—
310C4⋊C4—2442no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C22⋊C4—2442no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C42—41no—
320C42—612no—
321C42—612no—
322C42—1221no—
323C42—1222no—
324C42—1221no—
325C42—1222no—
326C42—1222no—
327C42—2441no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—2441no—
336C23—2441no—
337C23—2441no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23C2×C22⋊#692311yescharacteristic
343Q8—4882no—
344Q8—4882no—
345D4—82no—
346D4—4882no—
347D4—4882no—
348D4—4882no—
349D4—4882no—
350D4—4882no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—612no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—612no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—612no—
369C2×C4—612no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4D4×#692312yes—
380C8—4882no—
381C8—2444no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—4881no—
390C22—4881no—
391C22—4881no—
392C22—4881no—
393C22—4881no—
394C22—4881no—
395C22—4881no—
396C22#1684071411yescharacteristic
397C22#1684074811yescharacteristic
398C22#1684074911yescharacteristic
399C22#1684075211yescharacteristic
400C22#1684075311yescharacteristic
401C22#1684075411yescharacteristic
402C22#1684075411yescharacteristic
403C4—22no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4—612no—
408C4—612no—
409C4—612no—
410C4—612no—
411C4—4881no—
412C4—2441no—
413C4—2441no—
414C2—161no—
415C2—9761no—
416C2#2146134211yescharacteristic
417C2#2146134311yescharacteristic
418C2#2146135011yescharacteristic
419C2#2146135111yescharacteristic
420C2#2146195111yescharacteristic
421C2#2146196311yescharacteristic
422C2#2146198711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C22
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9710
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461735C2semidirect
2. #21461604C2semidirect
3. C2×#16840621C2semidirect
236. C61#9710semidirect

Maximal subgroups

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

subgroupindex
#214617352
#214616042
C2×#168406212
C2×#168406202
#214541732
#971061