Finite Group Databasebeta

← #29680615

Subgroups — #29680615

Order 15616.

842 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839714C211yesmaximal, characteristic
2#21448215C212yesmaximal
3#21445982C211yesmaximal, characteristic, complemented
4#21445980C211yesmaximal, characteristic, complemented
5#21445948C212yesmaximal, complemented
6#21445853C211yesmaximal, characteristic
7#21445851C212yesmaximal
8#21445849C211yesmaximal, characteristic
9#21445717C212yesmaximal, complemented
10#21445716C211yesmaximal, characteristic, complemented
11C61×#1776C211yesmaximal, characteristic, complemented
12C22×#11758189C2211yescharacteristic
13C22×#11758189C2211yescharacteristic
14C2×C42×D61C2211yescharacteristic
15C2×#14519711C2211yescharacteristic
16C2×#14519710C2212yes—
17C2×#14519709C2211yescharacteristic
18C2×#14519708C2212yes—
19#16839714C418yescomplemented
20C2×#14519688C2211yescharacteristic
21C2×C8×Dic61C2211yescharacteristic
22C2×C8×Dic61C2211yescharacteristic
23#16839629—22no—
24#16839627C2212yes—
25#16839627C2212yes—
26#16839624—22no—
27#16839619C2212yes—
28#16839619C2212yes—
29C4×C4⋊Dic61C2212yes—
30C4×#11758198—22no—
31#16839608—22no—
32#16839525C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C2×#107×C61C2211yescharacteristic
36C2×#106×C61C2211yescharacteristic
37C2×#97×C61C2211yescharacteristic
38C4×C4⋊C4×C61C2212yes—
39C61×#340C2212yes—
40C22×C4×D61D412yes—
41C22×C4×D61C2311yescharacteristic
42C2×#11758189—22no—
43C2×#11758189—22no—
44C2×#11758189—22no—
45C2×#11758189—22no—
46C2×#11758189C2×C414yes—
47C2×#11758189C2×C414yes—
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50C2×C4⋊Dic61C2312yes—
51C2×C4⋊Dic61C2312yes—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×C4×Dic61—22no—
55C2×C4×Dic61D412yes—
56C2×C4×Dic61C2311yescharacteristic
57C42×D61C2×C414yes—
58C42×D61Q814yes—
59#14519711—22no—
60#14519710—24no—
61#14519709—22no—
62#14519708—24no—
63#14519688C2×C414yes—
64C8×Dic61C2×C414yes—
65C8×Dic61C2×C414yes—
66#14519684—22no—
67#14519684—22no—
68#14519684—22no—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684—22no—
73C22×C8×C61C2311yescharacteristic
74C22×C8×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2312yes—
76C2×C4⋊C4×C61C2312yes—
77C2×C42×C61—22no—
78C2×C42×C61C2311yescharacteristic
79#107×C61—22no—
80#106×C61—22no—
81#97×C61C2×C414yes—
82#95×C61—22no—
83C23×D61C2×D411yescharacteristic
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2411yescharacteristic
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61C2×D412yes—
91C22×Dic61C2×D411yescharacteristic
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—42no—
97C2×C4×D61—42no—
98C2×C4×D61C4⋊C418yes—
99C2×C4×D61C22×C412yes—
100C2×C4×D61C2×Q812yes—
101C2×C8×C61—21no—
102C2×C8×C61—21no—
103C2×C8×C61C22×C412yes—
104C2×C8×C61C22×C412yes—
105C2×C8×C61#4112yes—
106C2×C8×C61#4112yes—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C42×C61—44no—
112C42×C61C22×C412yes—
113C42×C61C2×Q812yes—
114#11758198—44no—
115#11758198—44no—
116C4⋊Dic61—22no—
117C4⋊Dic61—22no—
118C4⋊Dic61—22no—
119C4⋊Dic61—22no—
120C4×Dic61—44no—
121C4×Dic61C4⋊C418yes—
122C4×Dic61C22×C412yes—
123C4×Dic61C2×Q812yes—
124C2×#92418—21no—
125C2×#92418—21no—
126C2×#92418C22×C412yes—
127C2×#92418C22×C412yes—
128C2×#92418#4112yes—
129C2×#92418#4112yes—
130#11758189—28no—
131#11758189—28no—
132C23×C61C22×D411yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—22no—
136C22×D61—22no—
137C22×D61C2×C4⋊C412yes—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61C2×C4⋊C414yes—
147C2×C4×C61C23×C411yescharacteristic
148C2×C4×C61C22×Q811yescharacteristic
149C2×C4×C61C2×#4111yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61C2×C4⋊C412yes—
164C2×Dic61C2×C4⋊C414yes—
165C4×D61—22no—
166C4×D61—22no—
167C4×D61—22no—
168C4×D61—22no—
169C4×D61—48no—
170C8×C61—22no—
171C8×C61#11714yes—
172#92418—22no—
173#92418#11714yes—
174#20825—611nomaximal, Sylow
175C22×C61C22×C4⋊C411yescharacteristic
176C22×C61#53211yescharacteristic
177C22×C61#53211yescharacteristic
178C22×C61#57512yes—
179C22×C61#57512yes—
180C2×D61—24no—
181C2×D61—22no—
182C2×D61—42no—
183C2×D61—41no—
184C2×D61—41no—
185C4×C61—24no—
186C4×C61—82no—
187C4×C61—82no—
188C4×C61C2×#11711yescharacteristic
189C4×C61C2×#11711yescharacteristic
190C4×C61#53311yescharacteristic
191C4×C61#53311yescharacteristic
192Dic61—24no—
193Dic61—22no—
194Dic61—82no—
195Dic61—82no—
196C2×C4×M16—611no—
197#2283—612no—
198#1846—611no—
199#1846—611no—
200#1835—612no—
201#1809—611no—
202#1809—611no—
203#1809—612no—
204#1776—611no—
205#1776—612no—
206#1776D6111yescharacteristic, complemented
207C2×C61#228511yescharacteristic
208C2×C61#291112yes—
209C2×C61#318312yes—
210C2×C61#318311yescharacteristic
211C2×C61#318311yescharacteristic
212D61—42no—
213C22×M16—611no—
214C22×M16—611no—
215C22×C42—611no—
216C2×#107—611no—
217C2×#107—612no—
218C2×#107C2×D6111yescharacteristic
219C2×#106—611no—
220C2×#106—612no—
221C2×#106C2×D6111yescharacteristic
222C4×M16—618no—
223C2×#97—611no—
224C2×#97C2×D6111yescharacteristic
225C2×C4×C8—611no—
226C2×C4×C8—611no—
227#383—1222no—
228#382—612no—
229#382—612no—
230#381—1222no—
231#380—612no—
232#380—612no—
233C4×C4⋊C4—612no—
234C4×C4⋊C4C2×D6112yes—
235C4×C22⋊C4—1222no—
236#376—1222no—
237#340—612no—
238#340—612no—
239#340—612no—
240#340C2×D6112yes—
241C61#2082511yescharacteristic, complemented, Sylow
242C23×C4—611no—
243C23×C4—612no—
244C2×M16—614no—
245C2×M16—614no—
246C2×M16—1222no—
247C2×M16—1222no—
248C2×M16—1222no—
249C2×M16—1222no—
250C22×C8—611no—
251C22×C8—611no—
252C22×C8C22×D6111yescharacteristic
253C22×C8C22×D6111yescharacteristic
254C2×C4⋊C4—612no—
255C2×C4⋊C4—612no—
256C2×C4⋊C4C22×D6112yes—
257C2×C4⋊C4C22×D6112yes—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1222no—
260C2×C42—22no—
261C2×C42—612no—
262C2×C42—611no—
263C2×C42—614no—
264C2×C42—614no—
265C2×C42—1222no—
266C2×C42C22×D6111yescharacteristic
267#107—22no—
268#107—1224no—
269#107—1222no—
270#106—22no—
271#106—1224no—
272#106—1222no—
273#97—614no—
274#97C4×D6114yes—
275C4×C8—614no—
276C4×C8—614no—
277#95—22no—
278#95—1222no—
279#95—1222no—
280#95—1222no—
281#95—1222no—
282#95—1222no—
283#95—1222no—
284#95—1222no—
285C24—611no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—612no—
289C22×C4—611no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—618no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—2442no—
300C22×C4—2442no—
301C22×C4D4×D6112yes—
302C22×C4C23×D6111yescharacteristic
303M16—1228no—
304M16—1228no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—612no—
308C2×C8—612no—
309C2×C8—612no—
310C2×C8—612no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8C2×C4×D6112yes—
314C2×C8C2×C4×D6112yes—
315C2×C8#1175822212yes—
316C2×C8#1175822212yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—1222no—
322C4⋊C4—1222no—
323C4⋊C4—1222no—
324C4⋊C4—1222no—
325C22⋊C4—2444no—
326C22⋊C4—2444no—
327C42—44no—
328C42—612no—
329C42—612no—
330C42—618no—
331C42—2444no—
332C42C2×C4×D6112yes—
333C42Q8×D6112yes—
334C23—612no—
335C23—1222no—
336C23—1222no—
337C23—1221no—
338C23—1221no—
339C23C2×D4×D6111yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—612no—
349C2×C4—614no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4—2448no—
359C2×C4—2442no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4C4⋊C4×D6114yes—
366C2×C4C22×C4×D6111yescharacteristic
367C2×C4C2×Q8×D6111yescharacteristic
368C2×C4#41×D6111yescharacteristic
369C2×C4#41×D6111yescharacteristic
370C2×C4C2×#1175822211yescharacteristic
371C2×C4C2×#1175822211yescharacteristic
372C8—22no—
373C8—614no—
374C8—1222no—
375C8#1451973414yes—
376C22—1222no—
377C22—1224no—
378C22—2442no—
379C22—2441no—
380C22—2441no—
381C22C2×C4⋊C4×D6111yescharacteristic
382C22#1684030911yescharacteristic
383C22#1684030911yescharacteristic
384C22#1684055812yes—
385C22#1684056112yes—
386C4—24no—
387C4—82no—
388C4—82no—
389C4—1222no—
390C4—1224no—
391C4—4882no—
392C4—4882no—
393C4#117×D6111yescharacteristic
394C4C2×#1451973411yescharacteristic
395C4#1684031611yescharacteristic
396C4#1684031611yescharacteristic
397C2—2442no—
398C2#2144822311yescharacteristic
399C2#2145440612yes—
400C2#2145654412yes—
401C2#2145654611yescharacteristic
402C2#2145655111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#20825
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #21445982C2semidirect
4. #21445980C2semidirect
5. #21445948C2semidirect
9. #21445717C2semidirect
10. #21445716C2semidirect
11. C61×#1776C2semidirect
19. #16839714C4semidirect
206. #1776D61semidirect
241. C61#20825semidirect

Maximal subgroups

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

subgroupindex
C2×#168397142
#214482152
#214459822
#214459802
#214459482
#214458532
#214458512
#214458492
#214457172
#214457162
C61×#17762
#2082561