Finite Group Databasebeta

← #29834549

Subgroups — #29834549

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840198C211yesmaximal, characteristic, complemented
2C2×#16840178C211yesmaximal, characteristic, complemented
3#21455491C211yesmaximal, characteristic, complemented
4#21455463C211yesmaximal, characteristic, complemented
5#21447931C212yesmaximal, complemented
6#21447921C212yesmaximal, complemented
7#21447914C212yesmaximal, complemented
8#21447904C212yesmaximal, complemented
9#21447605C211yesmaximal, characteristic, complemented
10#21447527C211yesmaximal, characteristic, complemented
11C2×C61×#357C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840343—21no—
15#16840329—21no—
16#16840198—24no—
17C2×#14519721C2212yes—
18C2×#14519718C2212yes—
19#16840178—24no—
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yescomplemented
22C2×C16×D61C2212yescomplemented
23#16840012—22no—
24#16840009—22no—
25#16840004C2212yescomplemented
26#16839999C2212yescomplemented
27#16839995C2212yes—
28#16839995C2212yes—
29#16839800C2211yescharacteristic
30#16839767—21no—
31#16839761—21no—
32#16839589C2212yescomplemented
33#16839587C2212yescomplemented
34#16839567C2211yescharacteristic, complemented
35#16839567C2211yescharacteristic, complemented
36#16839564C2211yescharacteristic, complemented
37#16839564C2211yescharacteristic, complemented
38C22×D8×C61C2211yescharacteristic
39C22×C16×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#357C2212yes—
42C61×#357C2212yes—
43C2×#11758222—21no—
44C2×#11758222D411yescharacteristic
45#14519799—22no—
46#14519799—22no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50#14519783—22no—
51#14519783—22no—
52C2×#11758194D411yescharacteristic
53C2×#11758189D412yes—
54C2×C8×D61C2312yes—
55#14519773—41no—
56#14519764—41no—
57#14519757—22no—
58#14519757—41no—
59#14519752—22no—
60#14519752—41no—
61#14519751—42no—
62#14519748—22no—
63#14519748—22no—
64#14519746—42no—
65#14519736—21no—
66#14519721—24no—
67#14519718—24no—
68#14519717D414yescomplemented
69#14519716—24no—
70C2×#11758185C2312yes—
71C16×D61—24no—
72#14519711C2311yescharacteristic
73#14519711C2311yescharacteristic
74#14519709—21no—
75#14519700—22no—
76#14519698—22no—
77#14519698—22no—
78#14519697—22no—
79C22×D4×C61—21no—
80C2×D8×C61—22no—
81C2×D8×C61—21no—
82C2×D8×C61—22no—
83C2×D8×C61C2312yes—
84C22×C8×C61C2311yescharacteristic
85C2×C4⋊C4×C61—21no—
86C2×C16×C61—22no—
87C2×C16×C61C2311yescharacteristic
88C2×C16×C61C2311yescharacteristic
89#107×C61—21no—
90#107×C61C2312yes—
91C24×C61—41no—
92C2×D4×C61—22no—
93C2×D4×C61—42no—
94C2×D4×C61—41no—
95C2×D4×C61—42no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61—42no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61D812yes—
102#11758222—22no—
103#11758222—22no—
104#11758222—42no—
105#11758222—42no—
106C2×D244—21no—
107C2×D244—42no—
108C2×D244D811yescharacteristic
109C2×C4×D61—41no—
110C2×C4×D61C2×D412yes—
111C2×Dic122—21no—
112C2×Dic122D811yescharacteristic
113D8×C61—22no—
114D8×C61—22no—
115D8×C61—42no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D412yes—
118C2×C8×C61#4112yes—
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—22no—
121C4⋊C4×C61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—42no—
124C22⋊Dic61—42no—
125#11758198—42no—
126#11758198—42no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—42no—
130C4×Dic61—41no—
131#11758194—22no—
132C2×#92418C2×D412yes—
133Dic244—21no—
134Dic244—21no—
135D488—21no—
136D488—21no—
137#11758190—42no—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61—21no—
142C16×C61—21no—
143#11758185C2×D414yes—
144C23×C61—42no—
145C23×C61—82no—
146C23×C61—161no—
147C23×C61—161no—
148C23×C61C2×D811yescharacteristic
149C22×D61—41no—
150C22×D61C2×D812yes—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—82no—
154C2×C4×C61—42no—
155C2×C4×C61—161no—
156C2×C4×C61#12112yes—
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61#13512yes—
159C2×C4×C61C22×D411yescharacteristic
160C22⋊D61—24no—
161C22⋊D61—82no—
162C22⋊D61—162no—
163C22⋊D61—162no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61C2×D812yes—
168D244—22no—
169D244—41no—
170D244—41no—
171D244—162no—
172C4×D61—22no—
173C4×D61—22no—
174C4×D61—82no—
175Dic122—22no—
176Dic122—41no—
177Dic122—41no—
178C8×C61#12112yes—
179C8×C61C22×D411yescharacteristic
180C8×C61C2×#4111yescharacteristic
181#92418—22no—
182#33612—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61—82no—
188C22×C61—164no—
189C22×C61#46612yes—
190C22×C61C22×D811yescharacteristic
191C2×D61—22no—
192C2×D61—22no—
193C2×D61—22no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—82no—
198C4×C61#46612yes—
199C4×C61C2×#12111yescharacteristic
200C4×C61C2×#13511yescharacteristic
201Dic61—22no—
202Dic61—81no—
203Dic61—81no—
204Dic61—81no—
205C2×#509—611no—
206C2×#504—611no—
207#3061—611no—
208#3058—611no—
209#2220—612no—
210#2218—612no—
211#2217—612no—
212#2215—612no—
213#2159—611no—
214#2138—611no—
215C2×#357D6111yescharacteristic, complemented
216C2×C61—22no—
217C2×C61—162no—
218C2×C61C2×#46611yescharacteristic
219C2×C61#341511yescharacteristic
220C2×C61#341611yescharacteristic
221D61—42no—
222D61—161no—
223C2×#135—611no—
224C22×D8C2×D6111yescharacteristic
225C2×#131—611no—
226#537—1221no—
227#535—1221no—
228#509—1224no—
229C2×SD32—612no—
230C2×D16—612no—
231#504—1224no—
232C2×M32—611no—
233C2×M32—612no—
234C22×C16—612no—
235C22×C16C2×D6111yescharacteristic
236#469—1222no—
237#468—1222no—
238#467—612no—
239#466—612no—
240#466—612no—
241#466—612no—
242#427—611no—
243C2×#107C2×D6111yescharacteristic
244#418—1221no—
245#417—1221no—
246#367—612no—
247#366—612no—
248#358—611no—
249#358—611no—
250#357—611no—
251#357—611no—
252#357C2×D6112yes—
253#357C2×D6112yes—
254C61#3361211yescharacteristic, complemented, Sylow
255C2×#41—611no—
256C2×#41—1221no—
257C22×D4—21no—
258#135—1222no—
259#135—1222no—
260C2×Q16—611no—
261C2×SD16—1221no—
262C2×D8—22no—
263C2×D8—22no—
264C2×D8—21no—
265C2×D8—611no—
266C2×D8C22×D6112yes—
267#131—1222no—
268#131—1222no—
269C2×M16—611no—
270C2×M16—612no—
271C22×C8—612no—
272C22×C8C22×D6111yescharacteristic
273#127—2441no—
274#124—2441no—
275C22⋊Q8—1222no—
276C22⋊Q8—2441no—
277#121—1222no—
278#121—1222no—
279#121—1222no—
280#121—2441no—
281#121—2442no—
282#120—2442no—
283#117—1221no—
284C2×C4⋊C4—21no—
285SD32—1224no—
286D16—1224no—
287M32—614no—
288M32—1224no—
289C2×C16—22no—
290C2×C16—612no—
291C2×C16—1224no—
292C2×C16C22×D6111yescharacteristic
293C2×C16C22×D6111yescharacteristic
294#107—21no—
295#107—611no—
296#107—611no—
297#107C22×D6112yes—
298#106—1221no—
299#103—1222no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C24—41no—
304#41—1222no—
305#41—1222no—
306#41—2442no—
307#41—2442no—
308C2×Q8—611no—
309C2×Q8—1221no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—612no—
315C2×D4—611no—
316C2×D4—1221no—
317C2×D4—2442no—
318C2×D4—2442no—
319C2×D4—2442no—
320C2×D4—2441no—
321C22×C4—41no—
322C22×C4—612no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325Q16—1221no—
326Q16—1221no—
327SD16—2442no—
328D8—22no—
329D8—22no—
330D8—42no—
331D8—1221no—
332D8—1221no—
333M16—1222no—
334M16—1224no—
335C2×C8—612no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8#1175822212yes—
339C2×C8D4×D6112yes—
340C2×C8D4×D6111yescharacteristic
341C2×C8C23×D6111yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—2442no—
347C22⋊C4—2442no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2441no—
352C42—2441no—
353C16—21no—
354C16—21no—
355C16—614no—
356C23—42no—
357C23—82no—
358C23—161no—
359C23—161no—
360C23—612no—
361C23—2441no—
362C23D8×D6111yescharacteristic
363Q8—1222no—
364Q8—2441no—
365Q8—2441no—
366D4—42no—
367D4—42no—
368D4—82no—
369D4—1224no—
370D4—1222no—
371D4—9762no—
372D4—9762no—
373D4—9762no—
374D4—4882no—
375D4—2441no—
376D4—2441no—
377C2×C4—42no—
378C2×C4—161no—
379C2×C4—612no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—4882no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4#1451975012yes—
387C2×C4D8×D6111yescharacteristic
388C2×C4#1451979812yes—
389C2×C4C2×D4×D6111yescharacteristic
390C8—1222no—
391C8#1451975012yes—
392C8C2×D4×D6111yescharacteristic
393C8C2×#1175822211yescharacteristic
394C22—21no—
395C22—21no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—164no—
400C22—1222no—
401C22—1222no—
402C22—1222no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22#1683999712yes—
407C22C2×D8×D6111yescharacteristic
408C4—82no—
409C4—1222no—
410C4—4881no—
411C4—4881no—
412C4—4881no—
413C4#1683999712yes—
414C4C2×#1451975011yescharacteristic
415C4C2×#1451979811yescharacteristic
416C2—22no—
417C2—162no—
418C2—9761no—
419C2—2442no—
420C2C2×#1683999711yescharacteristic
421C2#2145895611yescharacteristic
422C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#357C2
socleC22×C61C22×D8
hypercenterC2×#357D61
hypocenterC61#33612
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. C2×#16840178C2semidirect
3. #21455491C2semidirect
4. #21455463C2semidirect
5. #21447931C2semidirect
6. #21447921C2semidirect
7. #21447914C2semidirect
8. #21447904C2semidirect
9. #21447605C2semidirect
10. #21447527C2semidirect
11. C2×C61×#357C2semidirect
21. C2×#14519716C22semidirect
22. C2×C16×D61C22semidirect
25. #16840004C22semidirect
26. #16839999C22semidirect
32. #16839589C22semidirect
33. #16839587C22semidirect
34. #16839567C22semidirect
35. #16839567C22semidirect
36. #16839564C22semidirect
37. #16839564C22semidirect
68. #14519717D4semidirect
215. C2×#357D61semidirect
254. C61#33612semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
C2×#168401782
#214554912
#214554632
#214479312
#214479212
#214479142
#214479042
#214476052
#214475272
C2×C61×#3572
#3361261