Finite Group Databasebeta

← #29652467

Subgroups — #29652467

Order 15616.

1170 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16840393C211yesmaximal, characteristic, complemented
2#21455079C212yesmaximal
3#21444792C214yesmaximal, complemented
4#21444758C214yesmaximal, complemented
5#21444351C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C61×#2406C211yesmaximal, characteristic
8C2×#14519807—22no—
9#16840393—24no—
10#16840393—22no—
11#16840393C2214yes—
12C2×#14519776—22no—
13C2×#14519768—22no—
14C2×#14519756—21no—
15C2×#14519756C2212yes—
16C2×#14519736—21no—
17#16839924—24no—
18#16839918C2214yes—
19#16839913C2214yes—
20#16839763—22no—
21#16839726C2212yes—
22#16839498C2218yes—
23#16839488C2214yes—
24C22×#41×C61C2211yescharacteristic
25C2×#123×C61—21no—
26C22×C4⋊C4×C61C2211yescharacteristic
27C61×#395C2212yes—
28C61×#394C2212yes—
29C61×#386—21no—
30C61×#385—22no—
31C61×#381C2211yescharacteristic
32C22×Dic122—21no—
33C22×Dic122D412yes—
34#14519807—48no—
35C2×#11758202—24no—
36C2×#11758200—24no—
37C2×#11758194—22no—
38#14519776—28no—
39#14519776—28no—
40#14519768—28no—
41#14519768—28no—
42#14519756—24no—
43#14519756—24no—
44#14519756—22no—
45#14519756—22no—
46#14519756D414yes—
47#14519756C2314yes—
48#14519736—24no—
49#14519736—24no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D412yes—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57C2×C4×Dic61—22no—
58#14519701—24no—
59#14519698—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yes—
62C2×#41×C61—22no—
63C2×#41×C61—22no—
64C2×#41×C61—44no—
65C22×Q8×C61C2311yescharacteristic
66C22×D4×C61—21no—
67C22×D4×C61C2311yescharacteristic
68C23×C4×C61—21no—
69C23×C4×C61D411yescharacteristic
70C23×C4×C61D411yescharacteristic
71C23×C4×C61C2311yescharacteristic
72#123×C61—44no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61C2312yes—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81#95×C61—21no—
82#95×C61—22no—
83#95×C61C2312yes—
84C24×C61—21no—
85C24×C61C2×D411yescharacteristic
86#41×C61—28no—
87#41×C61—44no—
88#41×C61—44no—
89#41×C61—44no—
90C2×Q8×C61—22no—
91C2×Q8×C61—42no—
92C2×D4×C61—24no—
93C2×D4×C61—22no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—24no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—22no—
117C22×Dic61—22no—
118C2×Dic122—24no—
119C2×Dic122—22no—
120C2×Dic122—44no—
121C2×Dic122—82no—
122C2×Dic122C2×D414yes—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—44no—
126C22⋊C4×C61—44no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—44no—
129C22⋊Dic61—24no—
130C22⋊Dic61—24no—
131#11758202—48no—
132#11758200—48no—
133C4⋊Dic61—24no—
134C4⋊Dic61—24no—
135C4⋊Dic61—24no—
136C4⋊Dic61—22no—
137C4⋊Dic61C2×D414yes—
138#11758196—28no—
139#11758196—24no—
140#11758196—24no—
141C4×Dic61—28no—
142#11758194—44no—
143C2×#92418—24no—
144C23×C61—22no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—42no—
150C23×C61—41no—
151C23×C61—42no—
152C23×C61#12011yescharacteristic
153C23×C61#12011yescharacteristic
154C23×C61C22×D411yescharacteristic
155Q8×C61—24no—
156Q8×C61—42no—
157D4×C61—24no—
158D4×C61—42no—
159D4×C61—44no—
160D4×C61—44no—
161C2×C4×C61—24no—
162C2×C4×C61—24no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—24no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—48no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—81no—
180C2×C4×C61#12012yes—
181C2×C4×C61#13714yescomplemented
182C2×C4×C61C22×D412yes—
183C2×Dic61—24no—
184C2×Dic61—24no—
185C2×Dic61—44no—
186C2×Dic61—44no—
187Dic122—44no—
188Dic122—82no—
189Dic122—84no—
190#92418—44no—
191#18143—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—44no—
200C22×C61—44no—
201C22×C61—42no—
202C22×C61—82no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61B(4,𝔽2)12yes—
206C22×C61C2×#12011yescharacteristic
207C22×C61C2×#13712yes—
208C4×C61—24no—
209C4×C61—22no—
210C4×C61—22no—
211C4×C61—42no—
212C4×C61—42no—
213C4×C61—82no—
214Dic61—44no—
215Dic61—44no—
216C2×#544—611no—
217#3007—612no—
218#2406D6111yescharacteristic
219#1605—614no—
220#1599—614no—
221#1511—612no—
222#1463—611no—
223C2×C61—22no—
224C2×C61—22no—
225C2×C61—41no—
226C2×C61—82no—
227C2×C61#300411yescharacteristic
228C2×C61#302011yescharacteristic
229C2×C61C2×B(4,𝔽2)11yescharacteristic
230C22×#41C2×D6111yescharacteristic
231C2×#137—1222no—
232#544—614no—
233#544—1224no—
234#544—1222no—
235C2×C4⋊Q8—1222no—
236C2×#125—1222no—
237C2×#123—21no—
238C2×C22⋊Q8—612no—
239C2×C22⋊Q8—1221no—
240C2×#117—1221no—
241C22×C4⋊C4C2×D6111yescharacteristic
242#452—1224no—
243#451—614no—
244#450—614no—
245#417—1222no—
246#407—612no—
247#395C2×D6112yes—
248#394C2×D6112yes—
249#386—21no—
250#385—22no—
251#381C2×D6111yescharacteristic
252#328—618no—
253#324—614no—
254C61#1814311yescharacteristic, complemented, Sylow
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—44no—
258C22×Q8—612no—
259C22×Q8—1221no—
260C22×Q8C22×D6111yescharacteristic
261C22×D4—21no—
262C22×D4C22×D6111yescharacteristic
263C23×C4—21no—
264C23×C4C22⋊D6111yescharacteristic
265C23×C4C22⋊D6111yescharacteristic
266C23×C4C22×D6111yescharacteristic
267#137—2448no—
268C2×Q16—1224no—
269C2×SD16—1224no—
270C2×M16—1222no—
271C4⋊Q8—1228no—
272C4⋊Q8—1228no—
273#125—1228no—
274#125—1228no—
275#123—44no—
276C22⋊Q8—614no—
277C22⋊Q8—614no—
278C22⋊Q8—1224no—
279C22⋊Q8—1222no—
280C22⋊Q8—1222no—
281C22⋊Q8—1224no—
282#117—1224no—
283#117—1224no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—612no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1222no—
292C2×C4⋊C4—1222no—
293C2×C4⋊C4C22⋊D6112yes—
294C2×C4⋊C4C22×D6112yes—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—1222no—
299C2×C42—1222no—
300#103—1224no—
301#102—1224no—
302C22⋊C8—614no—
303C22⋊C8—1222no—
304#95—21no—
305#95—22no—
306#95C22×D6112yes—
307C24—21no—
308C24C2×C22⋊D6111yescharacteristic
309#41—28no—
310#41—44no—
311#41—44no—
312#41—44no—
313C2×Q8—22no—
314C2×Q8—42no—
315C2×Q8—614no—
316C2×Q8—1224no—
317C2×Q8—1222no—
318C2×Q8—4882no—
319C2×Q8—2444no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—41no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—1222no—
339C22×C4—1222no—
340C22×C4D4×D6112yes—
341C22×C4D4×D6112yes—
342C22×C4C2×C22⋊D6112yes—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6111yescharacteristic
345C22×C4C23×D6111yescharacteristic
346Q16—2448no—
347SD16—2448no—
348M16—2444no—
349C2×C8—1224no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—614no—
354C4⋊C4—1228no—
355C4⋊C4—1224no—
356C4⋊C4—1224no—
357C4⋊C4—1224no—
358C4⋊C4—1224no—
359C4⋊C4—1222no—
360C4⋊C4—1224no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—1224no—
365C22⋊C4—1224no—
366C42—1228no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—42no—
375C23#1451974511yescharacteristic
376C23#1451974511yescharacteristic
377C23C22×C22⋊D6111yescharacteristic
378Q8—24no—
379Q8—42no—
380Q8—4882no—
381Q8—4884no—
382Q8—2444no—
383D4—24no—
384D4—42no—
385D4—44no—
386D4—44no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—48no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—81no—
406C2×C4—1224no—
407C2×C4—1224no—
408C2×C4—2444no—
409C2×C4—2444no—
410C2×C4#1451974512yes—
411C2×C4#1451980714yescomplemented
412C2×C4C2×D4×D6112yes—
413C8—2444no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—44no—
423C22—42no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22#1683994812yes—
428C22C2×#1451974511yescharacteristic
429C22C2×#1451980712yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—2444no—
437C4—2444no—
438C2—22no—
439C2—22no—
440C2—41no—
441C2—82no—
442C2#2145505211yescharacteristic
443C2#2145518211yescharacteristic
444C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2406C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
hypercenter#2406D61
hypocenterC61#18143
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. C2×#16840393C2semidirect
3. #21444792C2semidirect
4. #21444758C2semidirect
5. #21444351C2semidirect
6. #21444149C2semidirect
181. C2×C4×C61#137semidirect
254. C61#18143semidirect
411. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
#214550792
#214447922
#214447582
#214443512
#214441492
C61×#24062
#1814361