Finite Group Databasebeta

← #29571548

Subgroups — #29571548

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21446242C212yesmaximal, complemented
3#21445807C211yesmaximal, characteristic
4#21444392C212yesmaximal, complemented
5C61×#1869C211yesmaximal, characteristic
6#16840471—24no—
7#16840469—22no—
8D4×Dic122—22no—
9#16840393C412yescomplemented
10#16840388—21no—
11#16840388C412yescomplemented
12#16840319—21no—
13C2×#14519756—21no—
14C2×#14519756C2212yes—
15#16840237—21no—
16C2×#14519702—22no—
17C2×#14519692—21no—
18#16839622C2212yes—
19#16839529—21no—
20#16839525—22no—
21#16839500—24no—
22C23×C8×C61C2211yescharacteristic
23C2×C4×D4×C61C2211yescharacteristic
24C2×C22⋊C8×C61—21no—
25C2×C22⋊C8×C61C2211yescharacteristic
26C61×#336—21no—
27C22×Dic122—21no—
28C22×Dic122D412yes—
29#14519777—21no—
30#14519777—21no—
31#14519776—24no—
32#14519769—22no—
33#14519768—24no—
34#14519756—22no—
35#14519756—22no—
36#14519756—22no—
37#14519756—22no—
38#14519756—22no—
39#14519756—22no—
40#14519756C2×C414yes—
41#14519756D414yes—
42#14519754—24no—
43#14519754—24no—
44#14519754—24no—
45#14519754—24no—
46C4×Dic122—22no—
47C4×Dic122—22no—
48D4×Dic61—24no—
49D4×Dic61—24no—
50#14519736—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61D412yes—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—22no—
59#14519702—44no—
60#14519692—42no—
61#14519684—22no—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61—21no—
66C22×C8×C61—22no—
67C22×C8×C61—21no—
68C22×C8×C61—42no—
69C4×D4×C61—22no—
70C4×D4×C61—22no—
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C22⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61D411yescharacteristic
74C2×C42×C61D411yescharacteristic
75C22⋊C8×C61—22no—
76C22⋊C8×C61—42no—
77C24×C61C22⋊C411yescharacteristic
78C24×C61C2×D411yescharacteristic
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61SD1612yes—
82C2×D4×C61Q1612yes—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61—22no—
100C22×Dic61—42no—
101C2×Dic122—22no—
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—81no—
107C2×Dic122—81no—
108C2×Dic122C22⋊C414yes—
109C2×C8×C61—22no—
110C2×C8×C61—24no—
111C2×C8×C61—42no—
112C2×C8×C61—41no—
113C2×C8×C61—41no—
114C2×C8×C61—41no—
115C2×C8×C61—41no—
116C2×C8×C61—41no—
117C4⋊C4×C61—22no—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—22no—
120C42×C61—21no—
121C42×C61—21no—
122C22⋊Dic61—24no—
123C22⋊Dic61—24no—
124C22⋊Dic61—24no—
125C22⋊Dic61—42no—
126C22⋊Dic61—42no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133C4⋊Dic61—42no—
134C4⋊Dic61C22⋊C414yes—
135#11758196—24no—
136#11758196—24no—
137#11758196—24no—
138#11758196—24no—
139C4×Dic61—24no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61#10312yescomplemented
147C23×C61C2×C22⋊C411yescharacteristic
148C23×C61#12012yes—
149C23×C61C2×SD1611yescharacteristic
150C23×C61C2×Q1611yescharacteristic
151C23×C61#13712yes—
152D4×C61—24no—
153D4×C61—24no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—81no—
169C2×C4×C61#10312yescomplemented
170C2×C4×C61C2×C22⋊C412yes—
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61#12011yescharacteristic
173C2×Dic61—24no—
174C2×Dic61—24no—
175C2×Dic61—82no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178C2×Dic61—82no—
179C2×Dic61—82no—
180C2×Dic61—82no—
181C2×Dic61—82no—
182Dic122—42no—
183Dic122—42no—
184Dic122—44no—
185Dic122—82no—
186Dic122—82no—
187C8×C61—42no—
188C8×C61—81no—
189#8351—611nomaximal, Sylow
190C22×C61—22no—
191C22×C61—21no—
192C22×C61—22no—
193C22×C61—21no—
194C22×C61—22no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—42no—
198C22×C61—42no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—81no—
202C22×C61#37911yescharacteristic
203C22×C61C2×#10311yescharacteristic
204C22×C61#41911yescharacteristic
205C22×C61#44812yes—
206C22×C61#45112yes—
207C4×C61—22no—
208C4×C61—21no—
209C4×C61—22no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61B(4,𝔽2)11yescharacteristic
213C4×C61#45811yescharacteristic
214Dic61—44no—
215Dic61—44no—
216Dic61—162no—
217#3501—611no—
218#1892—612no—
219#1869D6111yescharacteristic
220#1799—611no—
221#1520—612no—
222C2×C61—22no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—22no—
226C2×C61—42no—
227C2×C61#178711yescharacteristic
228C2×C61#179011yescharacteristic
229C2×C61#179411yescharacteristic
230C23×C8C2×D6111yescharacteristic
231#554—1222no—
232#554—1224no—
233D4×Q8—1222no—
234#544—612no—
235#543—612no—
236#543—1221no—
237#533—1221no—
238C2×C22⋊Q8—612no—
239C2×C22⋊Q8—1221no—
240#518—1221no—
241C2×C4×D4C2×D6111yescharacteristic
242C2×#103—1222no—
243C2×#99—1221no—
244C2×C22⋊C8—21no—
245C2×C22⋊C8C2×D6111yescharacteristic
246#380—612no—
247#342—1221no—
248#340—1222no—
249#336—21no—
250#328—1224no—
251C61#835111yescharacteristic, complemented, Sylow
252C22×Q8—612no—
253C22×Q8—1221no—
254C22×D4D24411yescharacteristic
255C23×C4C22⋊D6111yescharacteristic
256C23×C4C22×D6111yescharacteristic
257C22×C8—22no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—42no—
261C4⋊Q8—1221no—
262C4⋊Q8—1221no—
263C4⋊Q8—1224no—
264#125—1222no—
265#125—1224no—
266C22⋊Q8—614no—
267C22⋊Q8—614no—
268C22⋊Q8—1224no—
269C22⋊Q8—1224no—
270C22⋊Q8—1224no—
271C22⋊Q8—1224no—
272C22⋊Q8—1222no—
273C22⋊Q8—1222no—
274C22⋊Q8—1222no—
275C22⋊Q8—1222no—
276C22⋊Q8—1222no—
277C22⋊Q8—1222no—
278C4×Q8—1222no—
279C4×Q8—1222no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—1224no—
283C4×D4—1224no—
284#117—1222no—
285C2×C4⋊C4—612no—
286C2×C4⋊C4—1222no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4—1222no—
289C2×C4⋊C4—1222no—
290C2×C4⋊C4C4×D6111yescharacteristic
291C2×C22⋊C4—1222no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1222no—
294C2×C22⋊C4C4×D6111yescharacteristic
295C2×C22⋊C4D24411yescharacteristic
296C2×C42C22⋊D6111yescharacteristic
297#103—2444no—
298#99—2442no—
299C22⋊C8—22no—
300C22⋊C8—42no—
301#95—1222no—
302C24#1175819811yescharacteristic
303C24C2×D24411yescharacteristic
304C2×Q8—614no—
305C2×Q8—1222no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C2×Q8—4881no—
309C2×Q8—4881no—
310C2×Q8—2442no—
311C2×Q8—2442no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4#1175819012yes—
315C2×D4Dic24412yes—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—1222no—
323C22×C4—1222no—
324C22×C4—1222no—
325C22×C4—1222no—
326C22×C4—2442no—
327C22×C4#1175819811yescharacteristic
328C22×C4#1175819811yescharacteristic
329C22×C4#1175819811yescharacteristic
330C22×C4C2×C4×D6111yescharacteristic
331C22×C4D4×D6112yes—
332C22×C4D4×D6112yes—
333C22×C4C2×C22⋊D6111yescharacteristic
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C4⋊C4—22no—
343C4⋊C4—614no—
344C4⋊C4—1224no—
345C4⋊C4—1222no—
346C4⋊C4—1224no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—1224no—
351C4⋊C4—1222no—
352C4⋊C4—1224no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—1224no—
358C22⋊C4—1224no—
359C22⋊C4—1224no—
360C22⋊C4—2442no—
361C22⋊C4—2442no—
362C42—21no—
363C42—21no—
364C42—1224no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#1451970212yescomplemented
372C23C2×#1175819811yescharacteristic
373C23#1451974712yes—
374C23C2×#1175819011yescharacteristic
375C23C2×Dic24411yescharacteristic
376C23#1451981012yes—
377Q8—4882no—
378Q8—4882no—
379Q8—2444no—
380Q8—2442no—
381Q8—2442no—
382D4—24no—
383D4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—1224no—
400C2×C4—1224no—
401C2×C4—4882no—
402C2×C4—4882no—
403C2×C4—4882no—
404C2×C4—4882no—
405C2×C4—4882no—
406C2×C4—4882no—
407C2×C4—4882no—
408C2×C4#1451970212yescomplemented
409C2×C4C22⋊C4×D6112yes—
410C2×C4#1451974511yescharacteristic
411C2×C4#1451974511yescharacteristic
412C8—42no—
413C8—81no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22#1683961711yescharacteristic
427C22C2×#1451970211yescharacteristic
428C22#1683977211yescharacteristic
429C22#1683990312yes—
430C22#1683992012yes—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—9762no—
437C4—2444no—
438C4—2444no—
439C4#1683994811yescharacteristic
440C4#1683995111yescharacteristic
441C2—22no—
442C2—21no—
443C2—21no—
444C2—22no—
445C2—42no—
446C2#2144575511yescharacteristic
447C2#2144576611yescharacteristic
448C2#2144578211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1869C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#1869D61
hypocenterC61#8351
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. #21446242C2semidirect
4. #21444392C2semidirect
9. #16840393C4semidirect
11. #16840388C4semidirect
146. C23×C61#103semidirect
169. C2×C4×C61#103semidirect
251. C61#8351semidirect
371. C23#14519702semidirect
408. C2×C4#14519702semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214462422
#214458072
#214443922
C61×#18692
#835161