Finite Group Databasebeta

← #29654004

Subgroups — #29654004

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic
2#21456767C212yesmaximal, complemented
3#21444792C214yesmaximal, complemented
4#21444775C214yesmaximal
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#16840141—22no—
19#16840112—22no—
20#16839969—24no—
21#16839918C2218yes—
22#16839815C2212yes—
23#16839498C2218yes—
24#16839488C2212yes—
25#16839486C2212yes—
26C2×C4⋊Q8×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic
28C61×#400—21no—
29C61×#399—21no—
30C61×#395C2212yes—
31C61×#393C2212yes—
32C61×#389—21no—
33C61×#382C2211yescharacteristic
34C61×#380—21no—
35C2×#11758202—28no—
36#14519777—21no—
37#14519777—21no—
38#14519777D412yes—
39#14519777D412yes—
40#14519776—24no—
41#14519776—24no—
42#14519769—22no—
43#14519769—22no—
44#14519768—24no—
45#14519768—24no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756C2314yes—
50#14519754—24no—
51#14519754—24no—
52#14519754—24no—
53#14519754—24no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57C4×Dic122D414yes—
58#14519736—22no—
59#14519736—22no—
60#14519736—22no—
61#14519736—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61—22no—
66C4⋊#92418—22no—
67#14519701—28no—
68C22⋊#92418C2314yes—
69C4×#92418—22no—
70C22×Q8×C61C2312yes—
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—21no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61D411yescharacteristic
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—24no—
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×C61Q1614yescomplemented
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—42no—
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—
128#11758202—88no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61C2×D414yes—
137#11758196—24no—
138#11758196—24no—
139#11758196—24no—
140#11758196—24no—
141C4×Dic61—24no—
142C4×Dic61—24no—
143C2×#92418—24no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61C2×Q1612yes—
149C23×C61C22×D411yescharacteristic
150Q8×C61—48no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—24no—
154C2×C4×C61—24no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61#12012yes—
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12011yescharacteristic
170C2×C4×C61C2×Q1612yes—
171C2×C4×C61C2×Q1612yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C22×D412yes—
174C2×Dic61—24no—
175C2×Dic61—24no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178Dic122—44no—
179Dic122—82no—
180Dic122—82no—
181Dic122—82no—
182#92418—44no—
183#18303—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—81no—
191C22×C61C2×#12011yescharacteristic
192C22×C61C22×Q1611yescharacteristic
193C22×C61C2×#13711yescharacteristic
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—22no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—82no—
200C4×C61—82no—
201C4×C61#45112yes—
202C4×C61#45112yes—
203Dic61—44no—
204Dic61—44no—
205#3458—611no—
206#3209—612no—
207#2628D6111yescharacteristic
208#1605—614no—
209#1602—614no—
210#1518—612no—
211#1458—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61—42no—
215C2×C61C2×#45111yescharacteristic
216C2×C61#302211yescharacteristic
217C2×C61#302711yescharacteristic
218#544—612no—
219#544—1222no—
220#544—1222no—
221#544—1221no—
222#543—612no—
223#543—1221no—
224#533—612no—
225#533—1224no—
226#533—1221no—
227C2×C4⋊Q8C2×D6111yescharacteristic
228C2×#117—1221no—
229C2×#117C2×D6111yescharacteristic
230#494—1222no—
231#487—1222no—
232#462—1224no—
233#451—618no—
234#431—612no—
235#400—21no—
236#399—21no—
237#395C2×D6112yes—
238#393C2×D6112yes—
239#389—21no—
240#382C2×D6111yescharacteristic
241#380—21no—
242#328—618no—
243#324—612no—
244#323—612no—
245C61#1830311yescharacteristic, complemented, Sylow
246C22×Q8C22×D6112yes—
247C23×C4C22⋊D6111yescharacteristic
248C2×Q16—1228no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—612no—
252C4⋊Q8—612no—
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255C4⋊Q8—1224no—
256C4⋊Q8—1224no—
257#125—1222no—
258#125—1224no—
259#125—1224no—
260#125—1222no—
261C22⋊Q8—614no—
262C22⋊Q8—1224no—
263C22⋊Q8—1224no—
264C22⋊Q8—1224no—
265C22⋊Q8—1224no—
266C22⋊Q8—1222no—
267C22⋊Q8—1222no—
268C22⋊Q8—1222no—
269C4×Q8—614no—
270C4×Q8—1222no—
271C4×Q8—1222no—
272C4×Q8—1222no—
273#117—22no—
274#117—22no—
275#117—1222no—
276#117—1222no—
277#117—1222no—
278#117—1222no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C4⋊C4C22×D6111yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4C22⋊D6111yescharacteristic
290C2×C22⋊C4C22×D6111yescharacteristic
291C2×C42—1222no—
292C2×C42C22⋊D6111yescharacteristic
293C2×C42C22×D6111yescharacteristic
294C4⋊C8—1222no—
295#103—1228no—
296C22⋊C8—614no—
297C4×C8—1222no—
298#95—22no—
299#95—21no—
300#95—22no—
301#95C22×D6112yes—
302C24C2×C22⋊D6111yescharacteristic
303C2×Q8—24no—
304C2×Q8—82no—
305C2×Q8—614no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C2×Q8—1222no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4#1175820214yescomplemented
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C23×D6111yescharacteristic
323Q16—4888no—
324C2×C8—1224no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—614no—
333C4⋊C4—1222no—
334C4⋊C4—1222no—
335C4⋊C4—1224no—
336C4⋊C4—1224no—
337C4⋊C4—1222no—
338C4⋊C4—1222no—
339C4⋊C4—1224no—
340C4⋊C4—1222no—
341C4⋊C4—1222no—
342C4⋊C4—1224no—
343C4⋊C4—1222no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—82no—
347C22⋊C4—1224no—
348C22⋊C4—1224no—
349C42—21no—
350C42—21no—
351C42—1224no—
352C42—1224no—
353C42D4×D6112yes—
354C42D4×D6112yes—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23C2×#1175820212yes—
360C23C22×C22⋊D6111yescharacteristic
361Q8—48no—
362Q8—4882no—
363Q8—4882no—
364Q8—4882no—
365Q8—2444no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—1224no—
383C2×C4—1224no—
384C2×C4—4882no—
385C2×C4—4882no—
386C2×C4#1451974512yes—
387C2×C4#1451974511yescharacteristic
388C2×C4#1451974511yescharacteristic
389C2×C4C2×#1175820212yes—
390C2×C4C2×#1175820212yes—
391C2×C4#1451980712yes—
392C2×C4C2×D4×D6112yes—
393C8—2444no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×#1451974511yescharacteristic
402C22C22×#1175820211yescharacteristic
403C22C2×#1451980711yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—82no—
410C4—82no—
411C4—2444no—
412C4—2444no—
413C4#1683991712yes—
414C4#1683991712yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×#1683991711yescharacteristic
419C2#2145519711yescharacteristic
420C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2628C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#2628D61
hypocenterC61#18303
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. #21456767C2semidirect
3. #21444792C2semidirect
5. #21444382C2semidirect
99. C22×C4×C61Q16semidirect
245. C61#18303semidirect
316. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214567672
#214447922
#214447752
#214443822
#214441332
C61×#26282
#1830361