Finite Group Databasebeta

← #30601801

Subgroups — #30601801

Order 15616.

610 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 114 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#21461752C211yesmaximal, characteristic, complemented
2#21461604C211yesmaximal, characteristic, complemented
3C2×#16840622C212yesmaximal, complemented
4C2×#16840620C212yesmaximal
5#21454174C211yesmaximal, characteristic
6C2×#14519844—21no—
7C2×#14519844C2212yes—
8C2×#14519844C2212yes—
9C2×C4⋊#92420—21no—
10C2×#14519832—21no—
11#16840628—21no—
12#16840622—24no—
13#16840620—24no—
14C2×#14519775C2211yescharacteristic
15C2×#14519773C2211yescharacteristic
16C2×#14519766C411yescharacteristic
17C2×#14519765C411yescharacteristic
18C2×#14519764C411yescharacteristic
19#16839662—21no—
20#16839662—21no—
21#16839662—21no—
22#16839662—21no—
23C42×Dic61C2211yescharacteristic
24C2×#124×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844—42no—
31#14519844D414yes—
32C4⋊#92420—42no—
33#14519832—42no—
34C22×Dic122C2×C411yescharacteristic
35C22×D244C2×C411yescharacteristic
36C22×C22⋊D61—21no—
37C22×C22⋊D61—21no—
38#14519775—22no—
39#14519775—22no—
40#14519773—22no—
41#14519773—22no—
42#14519766—42no—
43#14519765—42no—
44#14519764—42no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×C4×Dic61C2×C411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56C2×C4×Dic61D411yescharacteristic
57C2×C4×Dic61D411yescharacteristic
58C2×C4×Dic61C2311yescharacteristic
59C22×Q8×C61C2×C411yescharacteristic
60C22×D4×C61C2×C411yescharacteristic
61#124×C61—42no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61C2×C411yescharacteristic
65C24×C61—21no—
66C23×D61—21no—
67C2×Q8×C61—22no—
68C2×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×D4×C61—22no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C22×C4×C61—21no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C2×C22⋊D61—42no—
78C2×C22⋊D61—41no—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C22×Dic61—21no—
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×D244—22no—
93C2×D244—41no—
94C2×D244—41no—
95C2×Dic122—22no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×#92420—24no—
99C2×#92420—42no—
100C2×#92420—42no—
101C2×#92420—42no—
102C2×#92420—42no—
103C2×#92420—42no—
104C22⋊C4×C61—42no—
105C22⋊C4×C61—42no—
106C42×C61—22no—
107C22⋊Dic61—42no—
108C22⋊Dic61—42no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112#11758198—42no—
113#11758196—42no—
114#11758196—41no—
115#11758196—41no—
116C4×Dic61—22no—
117C4×Dic61—22no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61—21no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124C4×Dic61C2×D412yes—
125C4×Dic61#4112yes—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61C2×C22⋊C411yescharacteristic
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141Q8×C61—82no—
142D4×C61—82no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61#10012yes—
154C2×C4×C61C4×D412yes—
155C22⋊D61—82no—
156C22⋊D61—82no—
157C22⋊D61—81no—
158C22⋊D61—81no—
159C22⋊D61—81no—
160C22⋊D61—81no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61#10012yes—
170C2×Dic61#12112yes—
171C2×Dic61#12411yescharacteristic
172C2×Dic61#12711yescharacteristic
173C2×Dic61#13512yes—
174C2×Dic61AGL(1,ℤ/8)11yescharacteristic
175C2×Dic61AGL(1,ℤ/8)11yescharacteristic
176C2×Dic61AGL(1,ℤ/8)11yescharacteristic
177C2×Dic61#13711yescharacteristic
178C2×Dic61#13711yescharacteristic
179C2×Dic61#13711yescharacteristic
180D244—82no—
181Dic122—82no—
182#92420—44no—
183#92420—82no—
184#9707—611nomaximal, Sylow
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39011yescharacteristic
193C22×C61C2×#10011yescharacteristic
194C22×C61C2×#10011yescharacteristic
195C22×C61#41611yescharacteristic
196C22×C61#41711yescharacteristic
197C22×C61#41711yescharacteristic
198C22×C61#41711yescharacteristic
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C2×D61—81no—
206C4×C61—22no—
207C4×C61—22no—
208C4×C61—41no—
209C4×C61—81no—
210Dic61—22no—
211Dic61—41no—
212Dic61—81no—
213Dic61#46312yes—
214Dic61#46412yes—
215Dic61#48911yescharacteristic
216Dic61#48911yescharacteristic
217Dic61#49611yescharacteristic
218Dic61#49711yescharacteristic
219#2858—611no—
220#1842—611no—
221#1767—611no—
222C2×#333—612no—
223C2×#331—612no—
224C2×C61—161no—
225C2×C61#155111yescharacteristic
226C2×C61#155111yescharacteristic
227C2×C61#156011yescharacteristic
228C2×C61#156011yescharacteristic
229C2×C61#195911yescharacteristic
230C2×C61#196511yescharacteristic
231C2×C61#198311yescharacteristic
232D61—161no—
233C2×C4⋊Q8—611no—
234C2×#127—611no—
235C2×#124—611no—
236C2×#124—611no—
237C2×#124—611no—
238C2×#124#692311yescharacteristic
239C2×C4⋊C8—612no—
240C2×C4⋊C8—612no—
241C2×C4⋊C8—1221no—
242C2×C4⋊C8—1221no—
243C2×#97—1221no—
244#390—1221no—
245#390—1221no—
246#390—1221no—
247#390—1221no—
248C43—611no—
249#336—1221no—
250#333—1224no—
251#331—1224no—
252C61#970711yescharacteristic, complemented, Sylow
253C22×Q8—611no—
254C22×Q8C2×#692311yescharacteristic
255C22×D4—611no—
256C22×D4—1221no—
257C22×D4—1221no—
258C22×D4C2×#692311yescharacteristic
259C22×C8—1222no—
260C22×C8—1222no—
261C4⋊Q8—1222no—
262C4⋊Q8—1222no—
263#127—1222no—
264#127—1222no—
265#124—42no—
266#124—2442no—
267#124—2442no—
268#124—2442no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C42—611no—
280C2×C42—611no—
281C2×C42—611no—
282C2×C42—611no—
283C2×C42—611no—
284C2×C42—611no—
285C2×C42C2×#692311yescharacteristic
286C4⋊C8—614no—
287C4⋊C8—1222no—
288C4⋊C8—1222no—
289C4⋊C8—1222no—
290C4⋊C8—2442no—
291C4⋊C8—2442no—
292#97—2442no—
293C24—21no—
294C24—1221no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—1222no—
299C2×Q8—2441no—
300C2×Q8—2441no—
301C2×D4—22no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—1222no—
305C2×D4—2441no—
306C2×D4—2441no—
307C2×D4—2442no—
308C2×D4—2441no—
309C2×D4—2441no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2441no—
316C2×D4—2441no—
317C22×C4—21no—
318C22×C4—611no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—1221no—
323C22×C4C22⋊#692311yescharacteristic
324C22×C4C22⋊#692311yescharacteristic
325C22×C4C22×#692311yescharacteristic
326C2×C8—1224no—
327C2×C8—2442no—
328C2×C8—2442no—
329C2×C8—2442no—
330C2×C8—2442no—
331C2×C8—2442no—
332C4⋊C4—2442no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—2442no—
338C22⋊C4—2442no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C42—22no—
344C42—612no—
345C42—612no—
346C42—1221no—
347C42—1221no—
348C42—1221no—
349C42—1222no—
350C42—1221no—
351C42—1221no—
352C42—1221no—
353C42—1222no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—2441no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23—2441no—
368C23C2×C22⋊#692311yescharacteristic
369Q8—82no—
370Q8—4882no—
371D4—82no—
372D4—4882no—
373D4—4881no—
374D4—4881no—
375D4—4881no—
376D4—4881no—
377D4—4882no—
378D4—4882no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—612no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—612no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—611no—
397C2×C4—611no—
398C2×C4—611no—
399C2×C4—612no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4#1451983712yes—
409C2×C4D4×#692312yes—
410C8—4882no—
411C8—2444no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—4881no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22#1684071411yescharacteristic
427C22C2×#1451983711yescharacteristic
428C22D61⋊#10011yescharacteristic
429C22#1684075211yescharacteristic
430C22#1684075311yescharacteristic
431C22#1684075311yescharacteristic
432C22#1684075311yescharacteristic
433C4—22no—
434C4—22no—
435C4—41no—
436C4—81no—
437C4—611no—
438C4—611no—
439C4—611no—
440C4—611no—
441C4—612no—
442C4—612no—
443C4—1222no—
444C4—4881no—
445C4—2441no—
446C2—161no—
447C2—9761no—
448C2#2146133811yescharacteristic
449C2#2146133911yescharacteristic
450C2#2146135611yescharacteristic
451C2#2146135711yescharacteristic
452C2#2146194911yescharacteristic
453C2#2146196111yescharacteristic
454C2#2146198711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C22
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#9707
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461752C2semidirect
2. #21461604C2semidirect
3. C2×#16840622C2semidirect
252. C61#9707semidirect

Maximal subgroups

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

subgroupindex
#214617522
#214616042
C2×#168406222
C2×#168406202
#214541742
#970761