Finite Group Databasebeta

← #30600665

Subgroups — #30600665

Order 15616.

512 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 94 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
1D4×#11758217C211yesmaximal, characteristic
2#21462012C211yesmaximal, characteristic, complemented
3#21461334C211yesmaximal, characteristic, complemented
4#21461144C211yesmaximal, characteristic
5#21460689C211yesmaximal, characteristic, complemented
6#21460664C211yesmaximal, characteristic, complemented
7C2×#16840050C211yesmaximal, characteristic
8C22×#11758217—21no—
9#16840797—21no—
10#16840793—22no—
11D4×#92420—21no—
12#16840779C2211yescharacteristic
13#16840765C2211yescharacteristic
14#16840765C2211yescharacteristic
15#16840732—21no—
16C4×#11758217C2211yescharacteristic
17#16840642—21no—
18#16840642—21no—
19#16840621—22no—
20D4⋊#92420—21no—
21D4⋊#92420—21no—
22C2×#14519776C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16840050—21no—
25#16840050—21no—
26#16840050—21no—
27#16840050—21no—
28C2×#14519709C411yescharacteristic
29C2×#14519708C411yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519698C411yescharacteristic
32C2×#102×C61C411yescharacteristic
33C2×#11758217—22no—
34C2×#11758217—21no—
35C2×#11758217—21no—
36C2×#11758217—42no—
37C2×#11758217—42no—
38C2×#11758217D411yescharacteristic
39C2×#11758217D411yescharacteristic
40C22×#92420—22no—
41#14519844—21no—
42#14519844—21no—
43#14519844D412yes—
44C4⋊#92420—21no—
45C4⋊#92420—21no—
46C22⋊#92420—22no—
47C22⋊#92420—22no—
48C22⋊#92420—21no—
49#14519832—21no—
50C4×#92420—21no—
51C4×#92420—21no—
52C22×Dic122—21no—
53C23×Dic61—21no—
54C22×#92418C2×C411yescharacteristic
55#14519776—21no—
56#14519776—21no—
57#14519776—21no—
58#14519776—21no—
59D4×Dic61—21no—
60D4×Dic61—21no—
61D4×Dic61D411yescharacteristic
62D4×Dic61D411yescharacteristic
63D4×Dic61D411yescharacteristic
64D4×Dic61D411yescharacteristic
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×#11758196—21no—
68C2×C22⋊Dic61—21no—
69C2×C4×Dic61C2311yescharacteristic
70#14519709—21no—
71#14519709—21no—
72#14519708—21no—
73#14519708—21no—
74#14519706—21no—
75#14519706—21no—
76#14519698—21no—
77#14519698—21no—
78C22×D4×C61C2×C411yescharacteristic
79C22×C8×C61C2×C411yescharacteristic
80C2×C4⋊C4×C61C2×C411yescharacteristic
81#102×C61—21no—
82#102×C61—21no—
83C24×C61—21no—
84C2×D4×C61—22no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C2×D4×C61C22⋊C411yescharacteristic
88C2×D4×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61C22×C411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61#4111yescharacteristic
103C2×Dic122—41no—
104C2×Dic122—41no—
105C2×Dic122—82no—
106#11758217—21no—
107#11758217—21no—
108#11758217—21no—
109#11758217—44no—
110#11758217—41no—
111#11758217—41no—
112#11758217—41no—
113#11758217—41no—
114#11758217SD1612yes—
115C2×#92420—21no—
116C2×#92420—21no—
117C2×#92420—22no—
118C2×#92420—44no—
119C2×#92420—41no—
120C2×#92420—41no—
121C2×C8×C61—21no—
122C2×C8×C61—41no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C22⋊Dic61—21no—
126C22⋊Dic61—21no—
127C22⋊Dic61—21no—
128C22⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61C22⋊C411yescharacteristic
133C4⋊Dic61C22⋊C411yescharacteristic
134#11758196—41no—
135#11758196—41no—
136C4×Dic61C2×D411yescharacteristic
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61#4111yescharacteristic
140C2×#92418—21no—
141C2×#92418—41no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—42no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—81no—
148C23×C61C4×D411yescharacteristic
149D4×C61—21no—
150D4×C61—21no—
151D4×C61—21no—
152D4×C61—42no—
153D4×C61#10112yes—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—81no—
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—81no—
175C2×Dic61#12011yescharacteristic
176C2×Dic61#12111yescharacteristic
177C2×Dic61#12111yescharacteristic
178C2×Dic61#12311yescharacteristic
179C2×Dic61C2×SD1611yescharacteristic
180C2×Dic61#13511yescharacteristic
181C2×Dic61AGL(1,ℤ/8)11yescharacteristic
182C2×Dic61#13711yescharacteristic
183Dic122—82no—
184Dic122—82no—
185#92420—22no—
186#92420—42no—
187#92420—41no—
188#92420—81no—
189#92420—81no—
190C8×C61—41no—
191#92418—41no—
192#9249—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—22no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—82no—
202C22×C61—82no—
203C22×C61#38611yescharacteristic
204C22×C61C4×SD1611yescharacteristic
205C22×C61#44011yescharacteristic
206C4×C61—21no—
207C4×C61—81no—
208C4×C61C2×#10111yescharacteristic
209C4×C61#41311yescharacteristic
210Dic61—21no—
211Dic61—21no—
212Dic61—41no—
213Dic61—41no—
214Dic61—81no—
215Dic61#45011yescharacteristic
216Dic61#45211yescharacteristic
217Dic61#46111yescharacteristic
218Dic61#46311yescharacteristic
219C2×#476—611no—
220D4×M16—611no—
221#1997—611no—
222#1549—611no—
223#1486—611no—
224#1292—611no—
225#1283—611no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—42no—
229C2×C61—81no—
230C2×C61#187711yescharacteristic
231C2×C61#190111yescharacteristic
232C2×C61#190311yescharacteristic
233C22×M16—1221no—
234C2×C4⋊Q8—611no—
235C2×C4×D4—611no—
236#476—1221no—
237#476—1221no—
238#476—1221no—
239#476—1221no—
240#436—1221no—
241#435—1222no—
242C8×D4—1221no—
243#431—611no—
244C2×#106—611no—
245C2×#106—611no—
246#423—611no—
247#423—611no—
248C2×C4⋊C8—611no—
249C2×#102—611no—
250C2×#102#692311yescharacteristic
251#407—1221no—
252C4×M16—611no—
253#343—1221no—
254#343—1221no—
255#332—1222no—
256D4⋊C8—1221no—
257D4⋊C8—1221no—
258C61#924911yescharacteristic, complemented, Sylow
259C22×Q8—1221no—
260C22×D4C2×#692311yescharacteristic
261C23×C4—1221no—
262C2×M16—611no—
263C2×M16—611no—
264C2×M16—1221no—
265C2×M16—1222no—
266C2×M16—1221no—
267C2×M16—2442no—
268C2×M16—2442no—
269C22×C8—611no—
270C22×C8—1222no—
271C22×C8C2×#692311yescharacteristic
272C4⋊Q8—1221no—
273C4⋊Q8—1221no—
274C4⋊Q8—1221no—
275C4⋊Q8—1221no—
276C4×D4—611no—
277C4×D4—611no—
278C4×D4—611no—
279C4×D4—611no—
280C4×D4—1221no—
281C4×D4—1221no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C2×#692311yescharacteristic
286C2×C22⋊C4—1221no—
287C2×C42—611no—
288#106—1221no—
289#106—1221no—
290#106—1221no—
291#106—1221no—
292C4⋊C8—612no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1221no—
297C4⋊C8—1221no—
298C4⋊C8—1221no—
299#102—21no—
300#102—21no—
301#102—1221no—
302#102—1221no—
303C22⋊C8—1222no—
304C22⋊C8—1222no—
305C22⋊C8—1221no—
306#97—1221no—
307C4×C8—1221no—
308C4×C8—1221no—
309C24—21no—
310C2×Q8—4882no—
311C2×Q8—2441no—
312C2×Q8—2441no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—41no—
316C2×D4C22⋊#692311yescharacteristic
317C2×D4C22⋊#692311yescharacteristic
318C22×C4—21no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—2441no—
328C22×C4—2441no—
329C22×C4—2441no—
330C22×C4—2441no—
331C22×C4C22×#692311yescharacteristic
332M16—612no—
333M16—1221no—
334M16—1221no—
335M16—1221no—
336M16—2444no—
337M16—2441no—
338M16—2441no—
339M16—2441no—
340M16—2441no—
341C2×C8—21no—
342C2×C8—41no—
343C2×C8—1221no—
344C2×C8—1221no—
345C2×C8—1222no—
346C2×C8—1221no—
347C2×C8—2444no—
348C2×C8—2441no—
349C2×C8—2441no—
350C2×C8—2441no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—611no—
354C4⋊C4—611no—
355C4⋊C4—1221no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—2441no—
359C4⋊C4—2441no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C22⋊C4—1221no—
364C42—611no—
365C42—611no—
366C42—611no—
367C42—611no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—81no—
374C23D4×#692311yescharacteristic
375Q8—4882no—
376Q8—4882no—
377D4—21no—
378D4—21no—
379D4—21no—
380D4—42no—
381D4#1451983812yes—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—611no—
387C2×C4—611no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—611no—
393C2×C4—611no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—4881no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4C2×C22⋊#692311yescharacteristic
410C2×C4D4×#692311yescharacteristic
411C8—41no—
412C8—1222no—
413C8—4881no—
414C8—4881no—
415C8—2442no—
416C8—2441no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—82no—
427C22—82no—
428C22#1684071011yescharacteristic
429C22SD16×#692311yescharacteristic
430C22#1684080111yescharacteristic
431C4—21no—
432C4—81no—
433C4—611no—
434C4—611no—
435C4—611no—
436C4—611no—
437C4—1221no—
438C4—1221no—
439C4—4881no—
440C4—2441no—
441C4—2441no—
442C4C2×#1451983811yescharacteristic
443C4#1684074911yescharacteristic
444C2—21no—
445C2—21no—
446C2—42no—
447C2—81no—
448C2#2146181211yescharacteristic
449C2#2146184611yescharacteristic
450C2#2146185011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#102×C61C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#102#6923
hypocenterC61#9249
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #21462012C2semidirect
3. #21461334C2semidirect
5. #21460689C2semidirect
6. #21460664C2semidirect
258. C61#9249semidirect

Maximal subgroups

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

subgroupindex
D4×#117582172
#214620122
#214613342
#214611442
#214606892
#214606642
C2×#168400502
#924961