Finite Group Databasebeta

← #29716295

Subgroups — #29716295

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#21455620C212yesmaximal
2#21448535C211yesmaximal, characteristic
3#21447166C212yesmaximal
4#21446977C212yesmaximal
5#21446801C212yesmaximal
6#21446212C212yesmaximal
7#21446034C212yesmaximal
8#21445815C211yesmaximal, characteristic
9C61×#3502C211yesmaximal, characteristic
10C2×#14519775C2211yescharacteristic
11C2×C4×Dic122C2211yescharacteristic
12#16840066—24no—
13#16840054—22no—
14#16840044—22no—
15#16839805C2214yes—
16#16839786C2212yes—
17#16839770C2214yes—
18C2×#14519696C2212yes—
19#16839669C2211yescharacteristic
20#16839667C2211yescharacteristic
21#16839646—22no—
22#16839643—22no—
23#16839627C2211yescharacteristic
24#16839608C2211yescharacteristic
25#16839531C2212yes—
26#16839521C2214yes—
27#16839521C2214yes—
28Q82×C61—22no—
29C61×#557—22no—
30C61×#557—24no—
31C61×#544—22no—
32C61×#544C2212yes—
33C61×#533—22no—
34C61×#533C2212yes—
35C61×#519C2212yes—
36C2×C4×Q8×C61C2211yescharacteristic
37C22×Dic122C2311yescharacteristic
38C2×#11758194C2314yes—
39#14519775—22no—
40#14519775Q814yescomplemented
41C4×Dic122—22no—
42C4×Dic122—22no—
43C2×C4⋊Dic61C2311yescharacteristic
44C2×#11758196—22no—
45C2×#11758196—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61C2311yescharacteristic
48C2×C4×Dic61C2311yescharacteristic
49#14519710—24no—
50#14519708—24no—
51C4⋊#92418—22no—
52C4⋊#92418—22no—
53#14519700—24no—
54#14519700—24no—
55#14519696—24no—
56#14519684—21no—
57#14519684—21no—
58#14519684—22no—
59#14519684—22no—
60#14519684—22no—
61C22×Q8×C61C2311yescharacteristic
62C4⋊Q8×C61—22no—
63C4⋊Q8×C61—22no—
64C4⋊Q8×C61—28no—
65C4⋊Q8×C61—24no—
66C4⋊Q8×C61D412yes—
67C4⋊Q8×C61D412yes—
68#125×C61—24no—
69#125×C61—28no—
70#125×C61—24no—
71#125×C61D414yes—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—24no—
74C22⋊Q8×C61—22no—
75C4×Q8×C61—24no—
76C4×Q8×C61—24no—
77C4×Q8×C61—24no—
78C4×Q8×C61—24no—
79C4×Q8×C61—22no—
80C4×Q8×C61—22no—
81C4×Q8×C61—22no—
82C4×Q8×C61—22no—
83#117×C61—24no—
84#117×C61C2314yes—
85C2×C4⋊C4×C61D412yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C42×C61D412yes—
88C2×C42×C61C2311yescharacteristic
89C2×Q8×C61—24no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61C2×Q812yes—
95C2×Q8×C61#4112yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C2×Dic122—22no—
106C2×Dic122—22no—
107C2×Dic122—21no—
108C2×Dic122C2×Q812yes—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—28no—
113C4⋊C4×C61—24no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61C2×D414yes—
120C4⋊C4×C61C2×D414yes—
121C22⋊C4×C61—24no—
122C42×C61—28no—
123C42×C61—22no—
124C42×C61—22no—
125C42×C61—24no—
126C42×C61C2×D412yes—
127C42×C61C2×D412yes—
128C4⋊Dic61—22no—
129#11758196—28no—
130#11758196—41no—
131#11758196—41no—
132C4×Dic61—21no—
133C4×Dic61—22no—
134C4×Dic61—42no—
135C4×Dic61C2×Q812yes—
136#11758194—24no—
137#11758194—24no—
138C2×#92418—24no—
139C23×C61C22×D411yescharacteristic
140Q8×C61—24no—
141Q8×C61—24no—
142Q8×C61—44no—
143Q8×C61—82no—
144Q8×C61—84no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—24no—
148C2×C4×C61—24no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61#12012yes—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61C22⋊Q814yes—
157C2×C4×C61C22×D412yes—
158C2×C4×C61C22×Q811yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×C4×C612−1+412yes—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172Dic122—24no—
173Dic122—42no—
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—44no—
186C4×C61—42no—
187C4×C61—44no—
188C4×C61#54812yes—
189C4×C61D4×Q812yes—
190Dic61—42no—
191Dic61—42no—
192Dic61—81no—
193Dic61—81no—
194#3502D6111yescharacteristic
195#3078—612no—
196#2327—611no—
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—22no—
209#557—22no—
210#557—24no—
211#544—22no—
212#544C2×D6112yes—
213#533—22no—
214#533C2×D6112yes—
215C2×C4⋊Q8—611no—
216#519C2×D6112yes—
217C2×C4×Q8—611no—
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#391—611no—
227#391—611no—
228#387—1222no—
229#387—1222no—
230#382—611no—
231#376—611no—
232#343—612no—
233#339—614no—
234#339—614no—
235C61#2431911yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C22×D6111yescharacteristic
238C2×M16—614no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—28no—
242C4⋊Q8—24no—
243C4⋊Q8—614no—
244C4⋊Q8—1222no—
245C4⋊Q8C22⋊D6112yes—
246C4⋊Q8C22⋊D6112yes—
247#125—24no—
248#125—28no—
249#125—24no—
250#125C22⋊D6114yes—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253C22⋊Q8—22no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—1222no—
263C4×Q8—1222no—
264#117—24no—
265#117C22×D6114yes—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4C22⋊D6112yes—
270C2×C4⋊C4C22×D6111yescharacteristic
271C2×C42—611no—
272C2×C42—611no—
273C2×C42—1221no—
274C2×C42C22⋊D6112yes—
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—1221no—
284#95—1221no—
285#95—1222no—
286#95—1222no—
287#95—1222no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—612no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—1222no—
297C2×Q8Q8×D6112yes—
298C2×Q8#1175822612yes—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
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—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—1228no—
322C4⋊C4—1222no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4C2×C22⋊D6114yes—
326C4⋊C4C2×C22⋊D6114yes—
327C22⋊C4—24no—
328C42—28no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—612no—
333C42—1221no—
334C42—1222no—
335C42—2442no—
336C42C2×C22⋊D6112yes—
337C42C2×C22⋊D6112yes—
338C23C22×C22⋊D6111yescharacteristic
339Q8—24no—
340Q8—24no—
341Q8—44no—
342Q8—82no—
343Q8—84no—
344Q8—1224no—
345Q8—2442no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2442no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4#1451974611yescharacteristic
366C2×C4#1451974611yescharacteristic
367C2×C4#1451974612yes—
368C2×C4#1451975314yes—
369C2×C4C22×C22⋊D6112yes—
370C2×C4C2×Q8×D6111yescharacteristic
371C2×C4C2×#1175822611yescharacteristic
372C2×C4#1451982512yes—
373C8—2444no—
374C22—21no—
375C22—21no—
376C22C2×#1451974611yescharacteristic
377C22C2×#1451975311yescharacteristic
378C22#1684033611yescharacteristic
379C4—21no—
380C4—21no—
381C4—22no—
382C4—24no—
383C4—44no—
384C4—42no—
385C4—44no—
386C4—4881no—
387C4—4881no—
388C4—2442no—
389C4—2442no—
390C4#1684042312yes—
391C4Q8×C22⋊D6112yes—
392C2—21no—
393C2—21no—
394C2#2144917811yescharacteristic
395C2#2145518311yescharacteristic
396C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#3502D61
hypocenterC61#24319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
40. #14519775Q8semidirect
235. C61#24319semidirect

Maximal subgroups

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

subgroupindex
#214556202
#214485352
#214471662
#214469772
#214468012
#214462122
#214460342
#214458152
C61×#35022
#2431961