Finite Group Databasebeta

← #29665362

Subgroups — #29665362

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic
3#21454900C211yesmaximal, characteristic, complemented
4#21445265C214yesmaximal, complemented
5#21445211C214yesmaximal, complemented
6#21444593C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic
9C2×C61×#462C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840165C2212yes—
13#16840110C2212yes—
14#16840075—22no—
15#16840039—22no—
16#16839984—22no—
17#16839972C2212yes—
18#16839962C2212yes—
19Q8×#92418—22no—
20Q8×#92418C2214yes—
21#16839815—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839786C2211yescharacteristic
25#16839761—21no—
26#16839514C2214yes—
27#16839509C2212yes—
28#16839508C2214yes—
29#16839505C2212yes—
30C22×Q16×C61—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C2×C4×Q8×C61C2211yescharacteristic
33C61×#462—22no—
34C61×#462C2214yes—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#103×C61—21no—
37C2×#11758222—21no—
38#14519808—44no—
39C2×#11758202—22no—
40C2×#11758201—22no—
41C2×#11758194—21no—
42#14519777C2311yescharacteristic
43#14519774C2311yescharacteristic
44#14519765—21no—
45#14519751—21no—
46#14519751—21no—
47C4×D244—21no—
48C4×D244—21no—
49#14519710—24no—
50C4⋊#92418—21no—
51C4⋊#92418—24no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—22no—
56#14519697—22no—
57#14519697—24no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C4×#92418—21no—
61C4×#92418—24no—
62C4×#92418C2312yes—
63C22×Q8×C61—21no—
64C22×Q8×C61D411yescharacteristic
65C2×Q16×C61—24no—
66C2×Q16×C61—44no—
67C22×C8×C61—21no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61C2312yes—
72C4×Q8×C61—21no—
73C4×Q8×C61—21no—
74C4×Q8×C61D412yes—
75C4×Q8×C61C2312yes—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C42×C61—21no—
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61—21no—
83C4⋊C8×C61C2312yes—
84#103×C61—24no—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61—22no—
88C2×Q8×C61—41no—
89C2×Q8×C61—42no—
90C2×Q8×C61—42no—
91C2×Q8×C61D814yes—
92C2×Q8×C61C2×D412yes—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—41no—
100#11758222—42no—
101#11758222—42no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×C4×D61—41no—
106C2×Dic122—21no—
107Q16×C61—48no—
108C2×C8×C61—22no—
109C2×C8×C61—42no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61C2×D412yes—
118C42×C61—22no—
119C42×C61—22no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124#11758202—44no—
125#11758201—44no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130#11758194—42no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418C2×D414yes—
134C23×C61#12111yescharacteristic
135C22×D61—41no—
136Q8×C61—22no—
137Q8×C61—24no—
138Q8×C61—42no—
139Q8×C61—42no—
140Q8×C61—82no—
141Q8×C61C2×D814yes—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—81no—
149C2×C4×C61—82no—
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61C2×D812yes—
154C2×C4×C61#13512yes—
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158C22⋊D61—82no—
159C2×Dic61—41no—
160D244—41no—
161D244—41no—
162D244—42no—
163D244—81no—
164C4×D61—82no—
165Dic122—41no—
166Dic122—41no—
167C8×C61—42no—
168#92418—24no—
169#92418—42no—
170#92418—42no—
171#19404—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61#46612yes—
175C22×C61C2×#12111yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—21no—
180C4×C61—24no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61—82no—
185C4×C61#46612yes—
186C4×C61C22×D811yescharacteristic
187C4×C61C2×#13511yescharacteristic
188C4×C61#57812yes—
189Dic61—81no—
190#3226—611no—
191#3037—611no—
192C2×#462D6111yescharacteristic, complemented
193#2982—611no—
194#1682—614no—
195#1674—614no—
196#1571—611no—
197#1546—611no—
198#1540—611no—
199C2×C61—22no—
200C2×C61#304411yescharacteristic
201C2×C61C2×#46611yescharacteristic
202C2×C61#306911yescharacteristic
203D61—161no—
204C2×#137—1221no—
205C22×Q16—21no—
206#532—611no—
207C2×C4⋊Q8C2×D6111yescharacteristic
208C2×C4×Q8C2×D6111yescharacteristic
209#500—612no—
210#486—612no—
211#480—1222no—
212#474—1222no—
213#464—1222no—
214#462—22no—
215#462—612no—
216#462C2×D6114yes—
217#460—612no—
218C8×Q8—614no—
219C8×Q8—1222no—
220#431—611no—
221#431—611no—
222#431—1222no—
223#423—611no—
224C2×C4⋊C8C2×D6111yescharacteristic
225#417—1221no—
226C2×#103—21no—
227#335—614no—
228#332—614no—
229#332—612no—
230#331—612no—
231C61#1940411yescharacteristic, complemented, Sylow
232C2×#41—1221no—
233C22×Q8—21no—
234C22×Q8C22⋊D6111yescharacteristic
235#137—2444no—
236C2×Q16—24no—
237C2×Q16—44no—
238C2×Q16—1222no—
239C2×SD16—1222no—
240C2×M16—1221no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—611no—
246C4⋊Q8C22×D6112yes—
247#127—611no—
248#124—1221no—
249#121—1221no—
250#121—1221no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D6112yes—
254C4×Q8C22×D6112yes—
255C4×D4—1221no—
256C4×D4—1221no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C42—21no—
262C2×C42C22×D6111yescharacteristic
263#107—1224no—
264C4⋊C8—21no—
265C4⋊C8—612no—
266C4⋊C8—612no—
267C4⋊C8—612no—
268C4⋊C8—1224no—
269C4⋊C8—1221no—
270C4⋊C8C22×D6112yes—
271#103—24no—
272#103—1222no—
273#102—1224no—
274#102—1222no—
275C22⋊C8—612no—
276C22⋊C8—1221no—
277C4×C8—612no—
278C4×C8—1224no—
279C4×C8—1221no—
280#41—2442no—
281#41—2442no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—1221no—
289C2×Q8#1175819914yes—
290C2×Q8C2×C22⋊D6112yes—
291C2×D4—1221no—
292C2×D4—1221no—
293C2×D4—1221no—
294C2×D4—2441no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—2441no—
299C22×C4D4×D6111yescharacteristic
300C22×C4#1175822411yescharacteristic
301C22×C4C2×C22⋊D6111yescharacteristic
302Q16—48no—
303Q16—2444no—
304SD16—2444no—
305M16—2442no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—614no—
309C2×C8—1222no—
310C2×C8—1222no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4C2×C22⋊D6112yes—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42—22no—
324C42—22no—
325C42D4×D6111yescharacteristic
326C42#1175822411yescharacteristic
327C42C2×C22⋊D6111yescharacteristic
328C42C23×D6111yescharacteristic
329C23—2441no—
330C23#1451974911yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—2441no—
337Q8—2441no—
338Q8C2×#1175819914yes—
339D4—4881no—
340D4—4882no—
341D4—2441no—
342D4—2441no—
343D4—2442no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—4882no—
353C2×C4—2441no—
354C2×C4#1451974911yescharacteristic
355C2×C4#1451974911yescharacteristic
356C2×C4#1451974911yescharacteristic
357C2×C4C2×#1175819912yes—
358C2×C4#1451979712yes—
359C2×C4C2×D4×D6111yescharacteristic
360C2×C4C22×C22⋊D6111yescharacteristic
361C2×C4C2×#1175822411yescharacteristic
362C8—42no—
363C8—1224no—
364C8—2442no—
365C8—2442no—
366C22—21no—
367C22—21no—
368C22—4881no—
369C22—4881no—
370C22—4881no—
371C22#1683999612yes—
372C22C2×#1451974911yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—82no—
379C4—4881no—
380C4#1683999612yes—
381C4C22×#1175819911yescharacteristic
382C4C2×#1451979711yescharacteristic
383C4#1684057512yes—
384C2—22no—
385C2—9761no—
386C2#2145534611yescharacteristic
387C2C2×#1683999611yescharacteristic
388C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#462C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#462D61
hypocenterC61#19404
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #21454900C2semidirect
4. #21445265C2semidirect
5. #21445211C2semidirect
6. #21444593C2semidirect
7. #21444491C2semidirect
9. C2×C61×#462C2semidirect
192. C2×#462D61semidirect
231. C61#19404semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214552912
#214549002
#214452652
#214452112
#214445932
#214444912
#214444692
C2×C61×#4622
#1940461