Finite Group Databasebeta

← #30601790

Subgroups — #30601790

Order 15616.

602 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 118 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
2C4×#14519844C211yesmaximal, characteristic, complemented
3C2×#16840621C212yesmaximal, complemented
4C2×#16840620C212yesmaximal
5#21454173C211yesmaximal, characteristic
6C2×#14519844C2212yes—
7C2×#14519844C2212yes—
8C2×C4×#92420—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—
31C4×#92420—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—42no—
42#14519763—42no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56C2×C4×Dic61C2311yescharacteristic
57C22×D4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C42×C61C2×C411yescharacteristic
62C24×C61—21no—
63C23×D61—21no—
64C2×D4×C61—22no—
65C2×D4×C61—41no—
66C2×D4×C61—41no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C2×C22⋊D61—42no—
72C2×C22⋊D61—41no—
73C2×C22⋊D61—41no—
74C2×C22⋊D61—41no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—42no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×D244—22no—
85C2×D244—41no—
86C2×D244—41no—
87C2×Dic122—42no—
88C2×Dic122—41no—
89C2×Dic122—41no—
90C2×Dic122—42no—
91C2×#92420—24no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C2×#92420—42no—
97C4⋊C4×C61—22no—
98C22⋊C4×C61—42no—
99C22⋊C4×C61—42no—
100C42×C61—41no—
101C22⋊Dic61—42no—
102C22⋊Dic61—42no—
103#11758198—42no—
104#11758198—42no—
105#11758198—42no—
106#11758198—42no—
107C4⋊Dic61—22no—
108#11758196—41no—
109#11758196—41no—
110C4×Dic61—22no—
111C4×Dic61—22no—
112C4×Dic61—22no—
113C4×Dic61—41no—
114C4×Dic61D812yes—
115C4×Dic61SD1612yes—
116C4×Dic61C2×D412yes—
117C4×Dic61#4112yes—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133D4×C61—82no—
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—21no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61#10212yes—
145C2×C4×C61C4×D412yes—
146C22⋊D61—82no—
147C22⋊D61—82no—
148C22⋊D61—82no—
149C22⋊D61—82no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61#10212yes—
159C2×Dic61#12112yes—
160C2×Dic61#12411yescharacteristic
161C2×Dic61#12711yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×SD1611yescharacteristic
164C2×Dic61#13512yes—
165C2×Dic61#13512yes—
166C2×Dic61AGL(1,ℤ/8)12yes—
167D244—82no—
168Dic122—82no—
169Dic122—82no—
170#92420—44no—
171#92420—82no—
172#9696—611nomaximal, Sylow
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61#39011yescharacteristic
181C22×C61#41311yescharacteristic
182C22×C61#41311yescharacteristic
183C22×C61C2×#10211yescharacteristic
184C22×C61#41611yescharacteristic
185C22×C61#41611yescharacteristic
186C22×C61#41811yescharacteristic
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—41no—
197C4×C61—81no—
198Dic61—22no—
199Dic61—41no—
200Dic61—81no—
201Dic61#45912yes—
202Dic61#46012yes—
203Dic61#48812yes—
204Dic61#49512yes—
205#2858—611no—
206#1831—611no—
207C4×C4⋊C8—611no—
208C2×#332—612no—
209C2×#331—612no—
210C2×C61—161no—
211C2×C61#154011yescharacteristic
212C2×C61#154011yescharacteristic
213C2×C61#155311yescharacteristic
214C2×C61#155311yescharacteristic
215C2×C61#195811yescharacteristic
216C2×C61#196711yescharacteristic
217C2×C61#198211yescharacteristic
218D61—161no—
219C2×C4⋊Q8—611no—
220C2×#127—611no—
221C2×#124—1221no—
222C2×#124—1221no—
223C2×C4⋊C8—612no—
224C2×C4⋊C8—612no—
225C2×C4×C8—1221no—
226#390—611no—
227#390—611no—
228#390—611no—
229#390—1221no—
230#390—1221no—
231#390#692311yescharacteristic
232C43—611no—
233#336—1221no—
234#336—1221no—
235#336—1221no—
236#332—1224no—
237#331—1224no—
238C61#969611yescharacteristic, complemented, Sylow
239C22×Q8—1221no—
240C22×D4—611no—
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4C2×#692311yescharacteristic
244C22×C8—1222no—
245C22×C8—1222no—
246C4⋊Q8—1222no—
247C4⋊Q8—1222no—
248#127—1222no—
249#127—1222no—
250#124—2442no—
251#124—2442no—
252#124—2444no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4C2×#692311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C42—611no—
265C2×C42—611no—
266C2×C42—611no—
267C2×C42—611no—
268C2×C42—611no—
269C2×C42—611no—
270C2×C42C2×#692311yescharacteristic
271C4⋊C8—614no—
272C4⋊C8—1222no—
273C4⋊C8—1222no—
274C4⋊C8—1222no—
275C4×C8—2442no—
276C24—21no—
277C24—1221no—
278C2×Q8—2441no—
279C2×Q8—2441no—
280C2×Q8—2442no—
281C2×Q8—2442no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—1222no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2441no—
289C2×D4—2441no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2442no—
295C2×D4—2442no—
296C22×C4—21no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—1221no—
302C22×C4C22⋊#692311yescharacteristic
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—1224no—
306C2×C8—2442no—
307C2×C8—2442no—
308C2×C8—2442no—
309C2×C8—2442no—
310C2×C8—2442no—
311C4⋊C4—22no—
312C4⋊C4—1222no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—41no—
324C42—612no—
325C42—612no—
326C42—612no—
327C42—612no—
328C42—1222no—
329C42—1222no—
330C42—1222no—
331C42—2441no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23—2441no—
345C23—2441no—
346C23C2×C22⋊#692311yescharacteristic
347Q8—4882no—
348Q8—4882no—
349D4—82no—
350D4—4882no—
351D4—4882no—
352D4—4882no—
353D4—4882no—
354D4—4882no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—612no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—611no—
369C2×C4—611no—
370C2×C4—612no—
371C2×C4—612no—
372C2×C4—612no—
373C2×C4—612no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451983912yes—
383C2×C4D4×#692312yes—
384C8—4882no—
385C8—2444no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—4881no—
394C22—4881no—
395C22—4881no—
396C22—4881no—
397C22—4881no—
398C22—4881no—
399C22—4881no—
400C22#1684071411yescharacteristic
401C22#1684074811yescharacteristic
402C22#1684074911yescharacteristic
403C22C2×#1451983911yescharacteristic
404C22#1684075211yescharacteristic
405C22#1684075211yescharacteristic
406C22#1684075411yescharacteristic
407C4—22no—
408C4—22no—
409C4—41no—
410C4—81no—
411C4—612no—
412C4—612no—
413C4—612no—
414C4—612no—
415C4—1222no—
416C4—4881no—
417C4—2441no—
418C2—161no—
419C2—9761no—
420C2#2146131611yescharacteristic
421C2#2146131711yescharacteristic
422C2#2146134211yescharacteristic
423C2#2146134311yescharacteristic
424C2#2146194811yescharacteristic
425C2#2146196311yescharacteristic
426C2#2146198611yescharacteristic
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#9696
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. C4×#14519844C2semidirect
3. C2×#16840621C2semidirect
238. C61#9696semidirect

Maximal subgroups

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

subgroupindex
#214617352
C4×#145198442
C2×#168406212
C2×#168406202
#214541732
#969661