Finite Group Databasebeta

← #29654072

Subgroups — #29654072

Order 15616.

888 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic
2#21456970C212yesmaximal, complemented
3#21444860C214yesmaximal, complemented
4#21444843C214yesmaximal
5#21444382C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic
7C61×#2628C211yesmaximal, characteristic
8#16840393—21no—
9#16840393C2212yes—
10#16840390—22no—
11#16840390—22no—
12#16840388—21no—
13#16840388C2212yes—
14#16840319—21no—
15#16840319C2212yes—
16#16840315—24no—
17C2×#14519736—21no—
18#16840165—22no—
19#16840114—22no—
20#16840097C2218yes—
21#16840054—24no—
22#16839815C2212yes—
23#16839498C2218yes—
24#16839488C2212yes—
25#16839486C2212yes—
26C2×C4⋊Q8×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C61×#400C2212yes—
29C61×#399C2212yes—
30C61×#395—21no—
31C61×#393—21no—
32C61×#389—21no—
33C61×#382C2211yescharacteristic
34C61×#380—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777D412yes—
38#14519777D412yes—
39#14519776—24no—
40#14519776—24no—
41#14519769—22no—
42#14519769—22no—
43#14519768—24no—
44#14519768—24no—
45#14519756—22no—
46#14519756—22no—
47#14519756—22no—
48#14519756C2314yes—
49#14519754—24no—
50#14519754—24no—
51#14519754—24no—
52#14519754—24no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C4×Dic122D414yes—
57#14519736—22no—
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C22⋊Dic61—22no—
64C2×C4×Dic61—22no—
65#14519710—28no—
66C4⋊#92418—22no—
67#14519700—28no—
68C22⋊#92418C2314yes—
69C4×#92418—22no—
70C22×Q8×C61—21no—
71C23×C4×C61D411yescharacteristic
72C4⋊Q8×C61—22no—
73C4⋊Q8×C61—22no—
74#117×C61—22no—
75#117×C61—22no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C4⋊C4×C61C2312yes—
81C2×C4⋊C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61D411yescharacteristic
84C2×C22⋊C4×C61C2311yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61C2311yescharacteristic
87#95×C61—22no—
88#95×C61—21no—
89#95×C61—22no—
90#95×C61C2312yes—
91C24×C61C2×D411yescharacteristic
92C2×Q8×C61—42no—
93C2×Q8×C61—82no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61Q1614yes—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—22no—
107C22×Dic61—22no—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122C2×D414yes—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—42no—
118C4⋊C4×C61—82no—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61—82no—
122C42×C61—21no—
123C42×C61—21no—
124C42×C61C2×D412yes—
125C42×C61C2×D412yes—
126C22⋊Dic61—24no—
127C22⋊Dic61—24no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61C2×D414yes—
136#11758196—24no—
137#11758196—24no—
138#11758196—24no—
139#11758196—24no—
140C4×Dic61—24no—
141C4×Dic61—24no—
142C2×#92418—24no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61C2×Q1612yes—
148C23×C61C22×D411yescharacteristic
149Q8×C61—84no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—24no—
153C2×C4×C61—24no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61#12012yes—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61C2×Q1612yes—
170C2×C4×C61C2×Q1612yes—
171C2×C4×C61#13712yes—
172C2×C4×C61C22×D412yes—
173C2×Dic61—24no—
174C2×Dic61—24no—
175C2×Dic61—82no—
176C2×Dic61—82no—
177Dic122—44no—
178Dic122—82no—
179Dic122—82no—
180Dic122—82no—
181#92418—44no—
182#18311—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—22no—
186C22×C61—42no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—81no—
190C22×C61C2×#12011yescharacteristic
191C22×C61C22×Q1611yescharacteristic
192C22×C61C2×#13711yescharacteristic
193C4×C61—22no—
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61—82no—
199C4×C61—82no—
200C4×C61#45112yes—
201C4×C61#45112yes—
202Dic61—44no—
203Dic61—44no—
204#3458—611no—
205#3230—612no—
206#2628D6111yescharacteristic
207#1617—614no—
208#1614—614no—
209#1518—612no—
210#1458—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61—42no—
214C2×C61C2×#45111yescharacteristic
215C2×C61#302111yescharacteristic
216C2×C61#302811yescharacteristic
217#544—612no—
218#544—1222no—
219#544—1222no—
220#544—1221no—
221#543—612no—
222#543—1221no—
223#533—612no—
224#533—1224no—
225#533—1221no—
226C2×C4⋊Q8C2×D6111yescharacteristic
227C2×#117—1221no—
228C2×#117C2×D6111yescharacteristic
229#500—1222no—
230#487—1222no—
231#484—618no—
232#477—1224no—
233#431—612no—
234#400C2×D6112yes—
235#399C2×D6112yes—
236#395—21no—
237#393—21no—
238#389—21no—
239#382C2×D6111yescharacteristic
240#380—21no—
241#328—618no—
242#324—612no—
243#323—612no—
244C61#1831111yescharacteristic, complemented, Sylow
245C22×Q8—21no—
246C23×C4C22⋊D6111yescharacteristic
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—612no—
250C4⋊Q8—612no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253C4⋊Q8—1224no—
254C4⋊Q8—1224no—
255#125—1222no—
256#125—1224no—
257#125—1224no—
258#125—1222no—
259C22⋊Q8—614no—
260C22⋊Q8—1224no—
261C22⋊Q8—1224no—
262C22⋊Q8—1224no—
263C22⋊Q8—1224no—
264C22⋊Q8—1222no—
265C22⋊Q8—1222no—
266C22⋊Q8—1222no—
267C4×Q8—614no—
268C4×Q8—1222no—
269C4×Q8—1222no—
270C4×Q8—1222no—
271#117—22no—
272#117—22no—
273#117—1222no—
274#117—1222no—
275#117—1222no—
276#117—1222no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4C22⋊D6111yescharacteristic
283C2×C4⋊C4C22×D6112yes—
284C2×C4⋊C4C22×D6111yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4C22⋊D6111yescharacteristic
288C2×C22⋊C4C22×D6111yescharacteristic
289C2×C42—1222no—
290C2×C42C22⋊D6111yescharacteristic
291C2×C42C22×D6111yescharacteristic
292#107—1228no—
293C4⋊C8—1222no—
294#103—1228no—
295C22⋊C8—614no—
296C4×C8—1222no—
297#95—22no—
298#95—21no—
299#95—22no—
300#95C22×D6112yes—
301C24C2×C22⋊D6111yescharacteristic
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8—614no—
305C2×Q8—1222no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—1222no—
314C22×C4—1222no—
315C22×C4#1175820214yes—
316C22×C4C2×C22⋊D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321C22×C4C23×D6111yescharacteristic
322C2×C8—1224no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C4⋊C4—614no—
331C4⋊C4—1222no—
332C4⋊C4—1222no—
333C4⋊C4—1224no—
334C4⋊C4—1224no—
335C4⋊C4—1222no—
336C4⋊C4—1222no—
337C4⋊C4—1224no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—1224no—
341C4⋊C4—1222no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—82no—
345C22⋊C4—1224no—
346C22⋊C4—1224no—
347C42—21no—
348C42—21no—
349C42—1224no—
350C42—1224no—
351C42D4×D6112yes—
352C42D4×D6112yes—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C2×#1175820212yes—
358C23C22×C22⋊D6111yescharacteristic
359Q8—84no—
360Q8—4882no—
361Q8—4882no—
362Q8—4882no—
363Q8—2444no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—1224no—
381C2×C4—1224no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4#1451974512yes—
385C2×C4#1451974511yescharacteristic
386C2×C4#1451974511yescharacteristic
387C2×C4C2×#1175820212yes—
388C2×C4C2×#1175820212yes—
389C2×C4#1451980712yes—
390C2×C4C2×D4×D6112yes—
391C8—2444no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22C2×#1451974511yescharacteristic
400C22C22×#1175820211yescharacteristic
401C22C2×#1451980711yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—82no—
408C4—82no—
409C4—2444no—
410C4—2444no—
411C4#1683991712yes—
412C4#1683991712yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2C2×#1683991711yescharacteristic
417C2#2145519011yescharacteristic
418C2#2145522411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2628C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#2628D61
hypocenterC61#18311
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #21456970C2semidirect
3. #21444860C2semidirect
5. #21444382C2semidirect
244. C61#18311semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214569702
#214448602
#214448432
#214443822
#214441332
C61×#26282
#1831161