Finite Group Databasebeta

← #30600404

Subgroups — #30600404

Order 15616.

708 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 112 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×#16840782C211yesmaximal, characteristic, complemented
2#21461675C211yesmaximal, characteristic, complemented
3C2×#16840620C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454162C211yesmaximal, characteristic
6#16840782—24no—
7C2×#14519844C2212yes—
8C2×#14519844C2212yes—
9C2×C22⋊#92420—21no—
10C2×C22⋊#92420—21no—
11C2×#14519832—21no—
12#16840628—21no—
13#16840620—24no—
14C2×#14519773C2211yescharacteristic
15C2×#14519751C411yescharacteristic
16C2×#14519749—21no—
17C2×#14519749—21no—
18C22×C4×Dic61C2211yescharacteristic
19#16839665C411yescharacteristic
20#16839663C411yescharacteristic
21#16839662—21no—
22#16839662—21no—
23#16839631C2211yescharacteristic
24C2×#121×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844D414yescomplemented
31C22⋊#92420—24no—
32C22⋊#92420—42no—
33#14519832—24no—
34C22×D244C2×C411yescharacteristic
35C22×C22⋊D61—21no—
36C22×C22⋊D61—21no—
37C22×C22⋊D61—21no—
38C23×Dic61D411yescharacteristic
39C23×Dic61D411yescharacteristic
40#14519773—22no—
41#14519773—22no—
42#14519751—42no—
43#14519749—44no—
44#14519749—44no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×C4×Dic61—22no—
54C2×C4×Dic61—22no—
55C2×C4×Dic61C2×C411yescharacteristic
56C2×C4×Dic61D411yescharacteristic
57C2×C4×Dic61D411yescharacteristic
58C2×C4×Dic61C2311yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61C2×C411yescharacteristic
61C23×C4×C61C2×C411yescharacteristic
62#121×C61—42no—
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C24×C61—21no—
66C24×C61C22⋊C411yescharacteristic
67C23×D61—21no—
68C2×D4×C61—22no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—42no—
72C2×D4×C61—42no—
73C2×D4×C61—42no—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—21no—
77C22×C4×C61—41no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—42no—
89C2×C22⋊D61—42no—
90C2×C22⋊D61—41no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61C22⋊C411yescharacteristic
101C22×Dic61C22⋊C411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C2×D244—22no—
105C2×D244—41no—
106C2×D244—41no—
107C2×#92420—24no—
108C2×#92420—42no—
109C2×#92420—42no—
110C2×#92420—42no—
111C2×#92420—42no—
112C2×#92420—42no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—42no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117#11758198—42no—
118#11758198—42no—
119#11758198—42no—
120C4⋊Dic61—41no—
121#11758196—42no—
122#11758196—42no—
123C4×Dic61—22no—
124C4×Dic61—21no—
125C4×Dic61—22no—
126C4×Dic61—21no—
127C4×Dic61C2×D412yes—
128C4×Dic61#4112yes—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61#10012yes—
143C23×C61C2×C22⋊C411yescharacteristic
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—41no—
151D4×C61—82no—
152D4×C61—84no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—44no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61C4×D412yes—
163C22⋊D61—84no—
164C22⋊D61—82no—
165C22⋊D61—82no—
166C22⋊D61—82no—
167C22⋊D61—82no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61#12011yescharacteristic
182C2×Dic61#12112yes—
183C2×Dic61#12311yescharacteristic
184C2×Dic61#13512yes—
185C2×Dic61AGL(1,ℤ/8)12yes—
186C2×Dic61AGL(1,ℤ/8)12yes—
187C2×Dic61AGL(1,ℤ/8)11yescharacteristic
188C2×Dic61#13711yescharacteristic
189D244—82no—
190#92420—44no—
191#92420—82no—
192#9142—611nomaximal, Sylow
193C22×C61—22no—
194C22×C61—22no—
195C22×C61—22no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61#38611yescharacteristic
208C22×C61C2×#10011yescharacteristic
209C22×C61#41311yescharacteristic
210C22×C61#41611yescharacteristic
211C22×C61#41711yescharacteristic
212C22×C61#41811yescharacteristic
213C22×C61#41811yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C4×C61—22no—
222C4×C61—41no—
223C4×C61—81no—
224Dic61—41no—
225Dic61—41no—
226Dic61—81no—
227Dic61#44912yes—
228Dic61#46812yes—
229Dic61#46912yes—
230Dic61#48212yes—
231C2×#432—611no—
232#2856—611no—
233#1797—611no—
234C2×#331—612no—
235#1296—612no—
236C2×C61—42no—
237C2×C61—161no—
238C2×C61#155111yescharacteristic
239C2×C61#155311yescharacteristic
240C2×C61#155511yescharacteristic
241C2×C61#155711yescharacteristic
242C2×C61#187511yescharacteristic
243C2×C61#189511yescharacteristic
244C2×C61#190011yescharacteristic
245D61—161no—
246C2×#127—611no—
247C2×#121—611no—
248C2×#121—1221no—
249C2×#121—1221no—
250C2×#121#692311yescharacteristic
251C22×C42—611no—
252#432—1224no—
253C2×C4⋊C8—612no—
254C2×C4⋊C8—612no—
255C2×C22⋊C8—1221no—
256C2×C22⋊C8—1221no—
257C2×#97—1221no—
258#390—611no—
259#390—611no—
260#390—1221no—
261#390—1221no—
262#384—611no—
263#336—1221no—
264#331—1224no—
265C61#914211yescharacteristic, complemented, Sylow
266C22×D4—21no—
267C22×D4—611no—
268C22×D4—1221no—
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4C2×#692311yescharacteristic
272C23×C4—611no—
273C23×C4—611no—
274C23×C4C2×#692311yescharacteristic
275C22×C8—1222no—
276C22×C8—1222no—
277#127—1222no—
278#127—1222no—
279#121—42no—
280#121—2442no—
281#121—2444no—
282#121—2444no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4C2×#692311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C42—611no—
294C2×C42—611no—
295C2×C42—611no—
296C2×C42—611no—
297C2×C42—1222no—
298C2×C42—1222no—
299C4⋊C8—614no—
300C4⋊C8—1222no—
301C4⋊C8—1222no—
302C4⋊C8—1222no—
303C22⋊C8—1224no—
304C22⋊C8—2442no—
305#97—1224no—
306C24—21no—
307C24—1221no—
308C24C22⋊#692311yescharacteristic
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—1222no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4—2442no—
319C2×D4—2442no—
320C2×D4—2441no—
321C2×D4—2442no—
322C2×D4—2441no—
323C2×D4—2442no—
324C2×D4—2441no—
325C2×D4—2441no—
326C2×D4—2442no—
327C2×D4—2442no—
328C2×D4—2442no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—611no—
337C22×C4—1221no—
338C22×C4—1221no—
339C22×C4—1222no—
340C22×C4—1221no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4C22⋊#692311yescharacteristic
347C22×C4C22×#692311yescharacteristic
348C2×C8—1224no—
349C2×C8—2442no—
350C2×C8—2442no—
351C2×C8—2442no—
352C2×C8—2442no—
353C2×C8—2442no—
354C4⋊C4—41no—
355C4⋊C4—2441no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C22⋊C4—42no—
359C22⋊C4—2442no—
360C22⋊C4—2442no—
361C22⋊C4—2442no—
362C22⋊C4—2442no—
363C22⋊C4—2442no—
364C42—612no—
365C42—612no—
366C42—1221no—
367C42—1222no—
368C42—1221no—
369C42—1222no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—2441no—
384C23—2441no—
385C23—2441no—
386C23—2441no—
387C23—2441no—
388C23—2441no—
389C23—2441no—
390C23#1451983712yes—
391C23C2×C22⋊#692311yescharacteristic
392D4—82no—
393D4—84no—
394D4—4882no—
395D4—4882no—
396D4—4882no—
397D4—4882no—
398D4—4882no—
399D4—4884no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—611no—
410C2×C4—612no—
411C2×C4—611no—
412C2×C4—612no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—612no—
416C2×C4—612no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1221no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4D4×#692312yes—
431C8—4882no—
432C8—2444no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—4881no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22—4881no—
454C22#1684071011yescharacteristic
455C22C2×#1451983711yescharacteristic
456C22#1684074911yescharacteristic
457C22#1684075211yescharacteristic
458C22#1684075311yescharacteristic
459C22#1684075411yescharacteristic
460C22#1684075411yescharacteristic
461C4—22no—
462C4—41no—
463C4—81no—
464C4—612no—
465C4—612no—
466C4—612no—
467C4—612no—
468C4—4881no—
469C4—2441no—
470C4—2441no—
471C2—42no—
472C2—161no—
473C2—9761no—
474C2#2146133811yescharacteristic
475C2#2146134311yescharacteristic
476C2#2146134611yescharacteristic
477C2#2146135111yescharacteristic
478C2#2146180711yescharacteristic
479C2#2146183511yescharacteristic
480C2#2146184411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454162C2
cosocleC2×C4×Dic61C23
hypercenterC2×#121#6923
hypocenterC61#9142
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16840782C2semidirect
2. #21461675C2semidirect
3. C2×#16840620C2semidirect
4. #21460697C2semidirect
30. #14519844D4semidirect
265. C61#9142semidirect

Maximal subgroups

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

subgroupindex
C2×#168407822
#214616752
C2×#168406202
#214606972
#214541622
#914261