Finite Group Databasebeta

← #30598227

Subgroups — #30598227

Order 15616.

1156 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 130 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#21461711C211yesmaximal, characteristic, complemented
2C2×#16840614C214yesmaximal, complemented
3#21460995C211yesmaximal, characteristic, complemented
4C2×#16840313C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×(C2×#6923)⋊C4—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840614—28no—
12C22×#11758226—21no—
13#16840313—28no—
14C2×#14519774C411yescharacteristic, complemented
15C2×#14519766—21no—
16C2×#14519752—21no—
17C2×#14519752C2212yes—
18C2×#14519739—21no—
19C2×#14519739C412yescomplemented
20C2×C42×D61C2211yescharacteristic
21C2×C4⋊Q8×C61C411yescharacteristic, complemented
22C23×#6923—22no—
23C22×#92420—22no—
24#14519834—24no—
25(C2×#6923)⋊C4—24no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—42no—
28C2×#11758226—28no—
29C2×#11758226—44no—
30C2×#11758226—42no—
31C2×#11758226—42no—
32C22×D244—22no—
33C22×D244—21no—
34C22×D244—21no—
35C22×D244C2×C412yes—
36C22×C4×D61—22no—
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40#14519774—24no—
41#14519766—28no—
42#14519752—28no—
43#14519752—24no—
44#14519752D418yes—
45#14519739—28no—
46#14519739—24no—
47#14519739—24no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C42×D61—24no—
54C22×Q8×C61—21no—
55C4⋊Q8×C61—24no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C412yes—
58C2×C42×C61C2×C411yescharacteristic
59C23×D61—22no—
60C23×D61C2×D411yescharacteristic
61C22×#6923—24no—
62C22×#6923—42no—
63C22×#6923—42no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C2×Q8×C61—24no—
69C2×Q8×C61—42no—
70C2×Q8×C61—42no—
71C22×C4×C61—22no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78#11758226—416no—
79#11758226—416no—
80C2×D244—24no—
81C2×D244—24no—
82C2×D244—28no—
83C2×D244—22no—
84C2×D244—42no—
85C2×D244—42no—
86C2×D244—42no—
87C2×D244—42no—
88C2×D244—42no—
89C2×D244—42no—
90C2×D244—44no—
91C2×D244—42no—
92C2×D244—42no—
93C2×D244C22⋊C414yes—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—28no—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×C4×D61—44no—
103C2×C4×D61—44no—
104C2×C4×D61D812yes—
105C2×C4×D61SD1612yes—
106C2×C4×D61C2×D414yes—
107C2×#92420—42no—
108C2×#92420—42no—
109C2×#92420—42no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61C22⋊C414yes—
113C42×C61—22no—
114#11758198—28no—
115#11758198—28no—
116C4×Dic61—22no—
117C4×Dic61—22no—
118C4×Dic61—22no—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—24no—
125C22×D61—42no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61#12012yes—
132C22×D61C2×D811yescharacteristic
133C22×D61C2×SD1611yescharacteristic
134C22×D61AGL(1,ℤ/8)12yes—
135C2×#6923—44no—
136C2×#6923—44no—
137C2×#6923—44no—
138C2×#6923—82no—
139C2×#6923—82no—
140C2×#6923—82no—
141C2×#6923—82no—
142Q8×C61—48no—
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—24no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61#10212yes—
153C2×C4×C61#10412yes—
154C2×C4×C61C2×C22⋊C412yes—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61#10212yes—
163C2×Dic61#12012yes—
164D244—44no—
165D244—44no—
166D244—48no—
167D244—48no—
168D244—48no—
169D244—48no—
170C4×D61—24no—
171C4×D61—24no—
172C4×D61—44no—
173C4×D61—416no—
174C4×D61—44no—
175#92420—82no—
176#8332—611nomaximal, Sylow
177C22×C61#37911yescharacteristic
178C22×C61#41011yescharacteristic
179C22×C61#41011yescharacteristic
180C22×C61C2×#10211yescharacteristic
181C22×C61#41811yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C22×C61#42111yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—22no—
191C2×D61—22no—
192C2×D61—44no—
193C2×D61—44no—
194C2×D61—44no—
195C2×D61—82no—
196C2×D61—82no—
197C2×D61—82no—
198C2×D61—82no—
199C2×D61#44712yes—
200C2×D61#45012yes—
201#6923—84no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—22no—
205C4×C61—44no—
206Dic61—22no—
207Dic61—22no—
208Dic61—41no—
209C2×#532—611no—
210#1817—611no—
211C2×#327—614no—
212#1434—611no—
213C2×C61#150211yescharacteristic
214C2×C61#150511yescharacteristic
215C2×C61#151411yescharacteristic
216C2×C61#151711yescharacteristic
217C2×C61#178211yescharacteristic
218C2×C61#178611yescharacteristic
219C2×C61#178911yescharacteristic
220D61—22no—
221D61—22no—
222D61—84no—
223C22×#41—1221no—
224#532—1228no—
225C2×C4⋊Q8#692311yescharacteristic, complemented
226C2×#127—611no—
227C2×#124—1221no—
228C2×#121—612no—
229C2×#121—1221no—
230C2×C4×D4—612no—
231C2×C4×D4—1221no—
232C22×C42—611no—
233C2×C22⋊C8—612no—
234C2×#95—612no—
235C2×#95—1221no—
236C2×#95—1221no—
237#336—1221no—
238#336—1221no—
239#327—1228no—
240C61#833211yescharacteristic, complemented, Sylow
241C2×#41—1228no—
242C2×#41—2442no—
243C2×#41—2442no—
244C2×#41—2444no—
245C22×Q8—21no—
246C22×D4—612no—
247C22×D4—1222no—
248C22×D4—1221no—
249C22×D4—1221no—
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—611no—
253C23×C4—1222no—
254C23×C4—1222no—
255C22×C8—1222no—
256C4⋊Q8—24no—
257#127—1224no—
258#124—1228no—
259#121—618no—
260#121—1224no—
261#121—1228no—
262C4×D4—1224no—
263C4×D4—1228no—
264C4×D4—1224no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×#692312yes—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1222no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—1224no—
273C2×C42C2×#692311yescharacteristic
274C22⋊C8—1224no—
275#95—1224no—
276#95—1224no—
277#95—2442no—
278C24—611no—
279C24—1222no—
280#41—24416no—
281#41—24416no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—614no—
286C2×D4—1228no—
287C2×D4—1224no—
288C2×D4—1224no—
289C2×D4—1222no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2444no—
297C2×D4—2442no—
298C2×D4—2442no—
299C22×C4—22no—
300C22×C4—611no—
301C22×C4—611no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—614no—
305C22×C4—611no—
306C22×C4—1222no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1228no—
310C22×C4—1222no—
311C22×C4—1224no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4—2444no—
316C22×C4—2444no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4C22⋊#692311yescharacteristic
324C22×C4C22⋊#692311yescharacteristic
325C22×C4C22×#692311yescharacteristic
326C2×C8—2442no—
327C2×C8—2442no—
328C2×C8—2442no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4C22⋊#692314yes—
332C22⋊C4—1228no—
333C22⋊C4—1228no—
334C42—22no—
335C42—1222no—
336C42—1222no—
337C42—1222no—
338C23—611no—
339C23—611no—
340C23—612no—
341C23—612no—
342C23—1221no—
343C23—1221no—
344C23—1221no—
345C23—1221no—
346C23—1224no—
347C23—2442no—
348C23—2442no—
349C23—2442no—
350C23—2442no—
351C23—2442no—
352C23—2442no—
353C23C2×C22⋊#692311yescharacteristic
354Q8—48no—
355D4—2444no—
356D4—2444no—
357D4—2448no—
358D4—2448no—
359D4—2448no—
360D4—2448no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—612no—
371C2×C4—612no—
372C2×C4—1224no—
373C2×C4—1224no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—4882no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4—24416no—
388C2×C4—2444no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4#1451983912yes—
392C2×C4#1451984212yes—
393C2×C4C2×C22⋊#692312yes—
394C8—4882no—
395C22—612no—
396C22—612no—
397C22—1221no—
398C22—1221no—
399C22—1222no—
400C22—1222no—
401C22—1221no—
402C22—1221no—
403C22—1222no—
404C22—1222no—
405C22—4882no—
406C22—4882no—
407C22—4882no—
408C22—4882no—
409C22—2444no—
410C22—2444no—
411C22—2444no—
412C22#1684070311yescharacteristic
413C22#1684074311yescharacteristic
414C22#1684074311yescharacteristic
415C22C2×#1451983911yescharacteristic
416C22#1684075411yescharacteristic
417C22C2×#1451984211yescharacteristic
418C22#1684076111yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—1222no—
424C4—1222no—
425C4—4884no—
426C4—2441no—
427C2—1222no—
428C2—1222no—
429C2—4884no—
430C2#2146120011yescharacteristic
431C2#2146121011yescharacteristic
432C2#2146124011yescharacteristic
433C2#2146125011yescharacteristic
434C2#2146165011yescharacteristic
435C2#2146165711yescharacteristic
436C2#2146166211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#16840313C2
cosocleC22×C4×D61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#8332
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21461711C2semidirect
2. C2×#16840614C2semidirect
3. #21460995C2semidirect
14. C2×#14519774C4semidirect
19. C2×#14519739C4semidirect
21. C2×C4⋊Q8×C61C4semidirect
225. C2×C4⋊Q8#6923semidirect
240. C61#8332semidirect

Maximal subgroups

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

subgroupindex
#214617112
C2×#168406142
#214609952
C2×#168403132
#833261