Finite Group Databasebeta

← #29716301

Subgroups — #29716301

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459629C211yesmaximal, characteristic
2#21455624C212yesmaximal
3#21447169C212yesmaximal
4#21446980C212yesmaximal
5#21446800C212yesmaximal
6#21446213C212yesmaximal
7#21446033C212yesmaximal
8#21445815C211yesmaximal, characteristic
9C61×#2327C211yesmaximal, characteristic
10Q8×Dic122—22no—
11#16840491—22no—
12#16840488—24no—
13#16840393—22no—
14#16840393C2212yes—
15#16840319—22no—
16#16840319C2212yes—
17#16840240C2212yes—
18#16840069—24no—
19#16840057—22no—
20#16840047—22no—
21#16839802C2214yes—
22#16839786C2212yes—
23#16839771C2214yes—
24C2×#14519696C2212yes—
25#16839531C2212yes—
26#16839522C2214yes—
27#16839522C2214yes—
28C2×C4⋊Q8×C61C2211yescharacteristic
29C2×C4×Q8×C61C2211yescharacteristic
30C2×C4×Q8×C61C2211yescharacteristic
31C61×#391C2211yescharacteristic
32C61×#391C2211yescharacteristic
33C61×#387—22no—
34C61×#387—22no—
35C61×#382C2211yescharacteristic
36C61×#376C2211yescharacteristic
37C2×#11758194C2314yes—
38#14519777—22no—
39#14519777—22no—
40#14519777D412yes—
41#14519777D412yes—
42#14519776—28no—
43#14519776—24no—
44#14519769—24no—
45#14519769D414yes—
46#14519768—28no—
47#14519768—24no—
48#14519756—22no—
49#14519756—24no—
50#14519756—22no—
51Q8×Dic61—24no—
52Q8×Dic61—24no—
53C4×Dic122—24no—
54C4×Dic122—24no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57#14519736—24no—
58#14519736C2314yes—
59#14519710—24no—
60#14519708—24no—
61C4⋊#92418—22no—
62C4⋊#92418—22no—
63#14519701—24no—
64#14519701—24no—
65#14519696—24no—
66C22×Q8×C61C2311yescharacteristic
67C22×Q8×C61C2311yescharacteristic
68C4⋊Q8×C61—22no—
69C4⋊Q8×C61Q814yescomplemented
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61C2311yescharacteristic
79C2×C42×C61—21no—
80C2×C42×C61D412yes—
81C2×C42×C61C2311yescharacteristic
82C2×C42×C61C2311yescharacteristic
83C2×C42×C61C2311yescharacteristic
84#95×C61—21no—
85#95×C61—21no—
86#95×C61—22no—
87#95×C61—22no—
88#95×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—21no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61C2×Q812yes—
95C2×Q8×C61C2×Q812yes—
96C2×Q8×C61#4112yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C2×Dic122—22no—
107C2×Dic122—24no—
108C2×Dic122—22no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—28no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C42×C61—22no—
116C42×C61—22no—
117C42×C61—21no—
118C42×C61—22no—
119C42×C61—42no—
120C42×C61C2×D412yes—
121C42×C61C2×D412yes—
122C42×C61C2×Q812yes—
123C22⋊Dic61—24no—
124C4⋊Dic61—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—24no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61C2×D414yes—
130C4⋊Dic61C2×D414yes—
131#11758196—28no—
132#11758196—24no—
133#11758196—24no—
134C4×Dic61—28no—
135C4×Dic61—24no—
136#11758194—24no—
137#11758194—24no—
138C2×#92418—24no—
139C23×C61C22×D411yescharacteristic
140Q8×C61—24no—
141Q8×C61—24no—
142Q8×C61—24no—
143Q8×C61—42no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12012yes—
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61C22⋊Q814yes—
165C2×C4×C61C22×D412yes—
166C2×C4×C61C22×Q811yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C2×C4×C612−1+412yes—
169C2×Dic61—24no—
170C2×Dic61—24no—
171Dic122—44no—
172Dic122—82no—
173Dic122—84no—
174#92418—44no—
175#24319—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61C2×#12011yescharacteristic
179C22×C61C2×C22⋊Q811yescharacteristic
180C22×C61#53611yescharacteristic
181C4×C61—24no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—22no—
185C4×C61—42no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—81no—
189C4×C61—81no—
190C4×C61#54812yes—
191C4×C61D4×Q812yes—
192Dic61—44no—
193Dic61—44no—
194#3502—611no—
195#3078—612no—
196#2327D6111yescharacteristic
197#2061—612no—
198#2028—612no—
199#1995—612no—
200#1886—612no—
201#1857—612no—
202#1801—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61#240911yescharacteristic
206C2×C61#302011yescharacteristic
207C2×C61#302211yescharacteristic
208Q82—1222no—
209#557—1224no—
210#557—1222no—
211#544—612no—
212#544—1222no—
213#533—612no—
214#533—1222no—
215C2×C4⋊Q8C2×D6111yescharacteristic
216#519—612no—
217C2×C4×Q8C2×D6111yescharacteristic
218C2×C4×Q8C2×D6111yescharacteristic
219#479—1224no—
220#477—1222no—
221#475—1222no—
222#428—614no—
223#423—612no—
224#419—614no—
225C2×#101—612no—
226#391C2×D6111yescharacteristic
227#391C2×D6111yescharacteristic
228#387—22no—
229#387—22no—
230#382C2×D6111yescharacteristic
231#376C2×D6111yescharacteristic
232#343—612no—
233#339—614no—
234#339—614no—
235C61#2431911yescharacteristic, complemented, Sylow
236C22×Q8C22×D6111yescharacteristic
237C22×Q8C22×D6111yescharacteristic
238C2×M16—614no—
239C4⋊Q8—22no—
240C4⋊Q8—612no—
241C4⋊Q8—612no—
242C4⋊Q8—1222no—
243C4⋊Q8—1222no—
244C4⋊Q8—1224no—
245C4⋊Q8—1228no—
246C4⋊Q8Dic12214yescomplemented
247#125—614no—
248#125—1224no—
249#125—1224no—
250#125—1228no—
251C22⋊Q8—1222no—
252C22⋊Q8—1222no—
253C22⋊Q8—1224no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—1224no—
259C4×Q8—1224no—
260C4×Q8—1224no—
261C4×Q8—1224no—
262C4×Q8—1222no—
263C4×Q8—1222no—
264#117—614no—
265#117—1224no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22⋊D6112yes—
269C2×C4⋊C4C22×D6111yescharacteristic
270C2×C4⋊C4C22×D6111yescharacteristic
271C2×C42—21no—
272C2×C42C22⋊D6112yes—
273C2×C42C22×D6111yescharacteristic
274C2×C42C22×D6111yescharacteristic
275C2×C42C22×D6111yescharacteristic
276#107—1224no—
277#106—1224no—
278C4⋊C8—1222no—
279C4⋊C8—1222no—
280#103—1224no—
281#103—1224no—
282#101—1224no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—1224no—
294C2×Q8—1222no—
295C2×Q8—1222no—
296C2×Q8C2×Dic12212yes—
297C2×Q8C2×Dic12212yes—
298C2×Q8#1175822212yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4C2×C22⋊D6112yes—
304C22×C4C2×C22⋊D6112yes—
305C22×C4C2×C22⋊D6111yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C23×D6111yescharacteristic
308M16—1224no—
309M16—1224no—
310C2×C8—1224no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—614no—
318C4⋊C4—614no—
319C4⋊C4—1228no—
320C4⋊C4—1224no—
321C4⋊C4—1224no—
322C4⋊C4—1224no—
323C4⋊C4—1224no—
324C4⋊C4—1224no—
325C4⋊C4—1222no—
326C4⋊C4—1222no—
327C22⋊C4—1224no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—1228no—
334C42—1224no—
335C42C2×Dic12212yes—
336C42D4×D6112yes—
337C42D4×D6112yes—
338C23C22×C22⋊D6111yescharacteristic
339Q8—24no—
340Q8—24no—
341Q8—24no—
342Q8—42no—
343Q8—4882no—
344Q8—4884no—
345Q8—2444no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1224no—
364C2×C4—1224no—
365C2×C4#1451974511yescharacteristic
366C2×C4#1451974511yescharacteristic
367C2×C4#1451974512yes—
368C2×C4#1451975614yes—
369C2×C4C2×D4×D6112yes—
370C2×C4C22×Dic12211yescharacteristic
371C2×C4C2×#1175822211yescharacteristic
372C2×C4#1451982412yes—
373C8—2444no—
374C22—21no—
375C22—21no—
376C22C2×#1451974511yescharacteristic
377C22C2×#1451975611yescharacteristic
378C22#1684033311yescharacteristic
379C4—21no—
380C4—21no—
381C4—22no—
382C4—24no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—81no—
387C4—81no—
388C4—2444no—
389C4—2444no—
390C4#1684042112yes—
391C4D4×Dic12212yes—
392C2—21no—
393C2—21no—
394C2#2144918411yescharacteristic
395C2#2145518211yescharacteristic
396C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2327C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#2327D61
hypocenterC61#24319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
69. C4⋊Q8×C61Q8semidirect
235. C61#24319semidirect
246. C4⋊Q8Dic122semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214556242
#214471692
#214469802
#214468002
#214462132
#214460332
#214458152
C61×#23272
#2431961