Finite Group Databasebeta

← #29834552

Subgroups — #29834552

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×#16840200C211yesmaximal, characteristic, complemented
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21455491C211yesmaximal, characteristic, complemented
4#21455468C211yesmaximal, characteristic, complemented
5#21447933C212yesmaximal, complemented
6#21447923C212yesmaximal, complemented
7#21447916C212yesmaximal, complemented
8#21447906C212yesmaximal, complemented
9#21447524C211yesmaximal, characteristic, complemented
10C2×#16839562C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×D488C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840343—21no—
16#16840331—21no—
17#16840200—24no—
18C2×#14519722C2212yes—
19C2×D976C2212yes—
20#16840180—24no—
21C2×#14519716C2212yescomplemented
22C22×#11758185C2211yescharacteristic
23C2×C16×D61C2212yescomplemented
24#16840016—22no—
25#16840009—22no—
26#16840004C2212yescomplemented
27#16839999C2212yes—
28#16839999C2212yes—
29#16839999C2212yescomplemented
30#16839766—21no—
31#16839761—21no—
32#16839565C2211yescharacteristic, complemented
33#16839565C2211yescharacteristic, complemented
34#16839562C2212yes—
35#16839562C2212yes—
36#16839562C2211yescharacteristic, complemented
37#16839562C2211yescharacteristic, complemented
38C2×M32×C61C2211yescharacteristic
39C61×#427C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#367C2212yescomplemented
42C61×#366C2212yescomplemented
43C2×#11758222—21no—
44C2×#11758222D411yescharacteristic
45C22×D244—21no—
46#14519799—22no—
47#14519799—22no—
48C2×Dic244C2311yescharacteristic
49C2×#11758190—21no—
50C2×D488—22no—
51C2×D488—22no—
52C2×D488—21no—
53C2×D488C2312yes—
54C2×D488C2311yescharacteristic
55#14519783—22no—
56#14519783—22no—
57C2×#11758194D411yescharacteristic
58C2×#11758189D412yes—
59C2×C8×D61C2312yes—
60#14519774—41no—
61#14519765—41no—
62#14519757—22no—
63#14519757—41no—
64#14519752—22no—
65#14519752—22no—
66#14519752—22no—
67#14519752—41no—
68#14519751—42no—
69#14519747—42no—
70#14519722—24no—
71D976—24no—
72#14519716—24no—
73C2×#11758185—22no—
74C2×#11758185C2311yescharacteristic
75C2×#11758185C2311yescharacteristic
76C16×D61—24no—
77#14519700—22no—
78#14519697—22no—
79#14519697—22no—
80#14519697—22no—
81C22×C8×C61C2311yescharacteristic
82#117×C61—21no—
83C2×C4⋊C4×C61—21no—
84M32×C61D414yescomplemented
85C2×C16×C61C2312yes—
86#107×C61—21no—
87#107×C61C2311yescharacteristic
88#107×C61C2311yescharacteristic
89#107×C61C2312yes—
90#106×C61—21no—
91C23×D61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C2×C22⋊D61D812yes—
96#11758222—22no—
97#11758222—22no—
98#11758222—42no—
99#11758222—42no—
100C2×D244—22no—
101C2×D244—21no—
102C2×D244—42no—
103C2×D244—42no—
104C2×D244—41no—
105C2×D244—42no—
106C2×D244—42no—
107C2×D244—42no—
108C2×D244D811yescharacteristic
109C2×C4×D61—41no—
110C2×C4×D61C2×D412yes—
111C2×Dic122—21no—
112C2×Dic122D811yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D412yes—
115C2×C8×C61#4112yes—
116C2×C8×C61C2411yescharacteristic
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61—41no—
123#11758198—42no—
124#11758198—42no—
125#11758198—42no—
126#11758198—42no—
127C4⋊Dic61—42no—
128#11758194—22no—
129C2×#92418C2×D412yes—
130Dic244—21no—
131Dic244—21no—
132D488—21no—
133D488—22no—
134D488—22no—
135D488—21no—
136D488—42no—
137#11758190—42no—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61C2×D414yes—
142#11758185—21no—
143#11758185—21no—
144C23×C61C2×D811yescharacteristic
145C22×D61—41no—
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—161no—
149C22×D61—161no—
150C22×D61C2×D812yes—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—161no—
155C2×C4×C61#12112yes—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61#13512yes—
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—24no—
160C22⋊D61—82no—
161C2×Dic61—41no—
162C2×Dic61C2×D812yes—
163D244—22no—
164D244—41no—
165D244—42no—
166D244—42no—
167D244—41no—
168D244—82no—
169D244—162no—
170D244—162no—
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#46612yes—
186C22×C61C22×D811yescharacteristic
187C2×D61—22no—
188C2×D61—22no—
189C2×D61—22no—
190C2×D61—81no—
191C2×D61—82no—
192C2×D61—82no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—82no—
196C2×D61—164no—
197C4×C61—81no—
198C4×C61—82no—
199C4×C61—81no—
200C4×C61#46612yes—
201C4×C61C2×#12111yescharacteristic
202C4×C61C2×#13511yescharacteristic
203Dic61—22no—
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#2159D6111yescharacteristic, complemented
214#2138—611no—
215C2×#357—611no—
216C2×C61—22no—
217C2×C61C2×#46611yescharacteristic
218C2×C61#341511yescharacteristic
219C2×C61#341611yescharacteristic
220D61—42no—
221D61—161no—
222D61—162no—
223C2×#135—611no—
224C22×D8—611no—
225C2×#131—611no—
226#537—1221no—
227#535—1221no—
228#509—1224no—
229C2×SD32—612no—
230C2×D16—612no—
231#504—1224no—
232C2×M32—612no—
233C2×M32C2×D6111yescharacteristic
234C22×C16—611no—
235C22×C16—612no—
236#469—1222no—
237#468—1222no—
238#467—612no—
239#466—612no—
240#466—612no—
241#466—612no—
242#427C2×D6111yescharacteristic
243C2×#107C2×D6111yescharacteristic
244#418—1221no—
245#417—1221no—
246#367C2×D6112yescomplemented
247#366C2×D6112yescomplemented
248#358—611no—
249#358—611no—
250#357—612no—
251#357—612no—
252#357—611no—
253#357—611no—
254C61#3361211yescharacteristic, complemented, Sylow
255C2×#41—611no—
256C2×#41—1221no—
257C22×D4—1221no—
258#135—1222no—
259#135—1222no—
260C2×Q16—611no—
261C2×SD16—1221no—
262C2×D8—612no—
263C2×D8—611no—
264C2×D8—1222no—
265C2×D8—1221no—
266C2×D8—1222no—
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—21no—
284C2×C4⋊C4—21no—
285SD32—1224no—
286D16—1224no—
287M32—1224no—
288M32D24414yescomplemented
289C2×C16—611no—
290C2×C16—611no—
291C2×C16—1222no—
292C2×C16—1224no—
293C2×C16C22×D6112yes—
294#107—21no—
295#107C22×D6112yes—
296#107C22×D6111yescharacteristic
297#107C22×D6111yescharacteristic
298#106—21no—
299#103—1222no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C24—2441no—
304#41—1222no—
305#41—1222no—
306#41—2442no—
307#41—2442no—
308C2×Q8—611no—
309C2×Q8—1221no—
310C2×D4—612no—
311C2×D4—611no—
312C2×D4—1222no—
313C2×D4—1221no—
314C2×D4—2442no—
315C2×D4—2442no—
316C2×D4—2442no—
317C2×D4—2441no—
318C2×D4—2442no—
319C2×D4—2441no—
320C2×D4—2442no—
321C22×C4—41no—
322C22×C4—612no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325Q16—1221no—
326Q16—1221no—
327SD16—2442no—
328D8—1221no—
329D8—1222no—
330D8—1222no—
331D8—1221no—
332D8—2442no—
333M16—1222no—
334M16—1224no—
335C2×C8—612no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8C2×D24412yes—
339C2×C8D4×D6111yescharacteristic
340C2×C8#1175822612yes—
341C2×C8C23×D6111yescharacteristic
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—2442no—
347C22⋊C4—41no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C42—41no—
353C16—1221no—
354C16—1221no—
355C16C2×D24414yes—
356C23—612no—
357C23—9761no—
358C23—9761no—
359C23—4882no—
360C23—2441no—
361C23—2442no—
362C23D8×D6111yescharacteristic
363Q8—1222no—
364Q8—2441no—
365Q8—2441no—
366D4—1224no—
367D4—1222no—
368D4—9762no—
369D4—9762no—
370D4—9762no—
371D4—4882no—
372D4—4882no—
373D4—2442no—
374D4—2441no—
375D4—2442no—
376D4—2441no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—161no—
381C2×C4—612no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—4882no—
385C2×C4—2441no—
386C2×C4#1451975212yes—
387C2×C4D8×D6111yescharacteristic
388C2×C4#1451980012yes—
389C2×C4C2×D4×D6111yescharacteristic
390C8—1222no—
391C8#1451975212yes—
392C8C22×D24411yescharacteristic
393C8C2×#1175822611yescharacteristic
394C22—21no—
395C22—21no—
396C22—1222no—
397C22—1222no—
398C22—1222no—
399C22—9764no—
400C22—4881no—
401C22—4882no—
402C22—4882no—
403C22—4881no—
404C22—4881no—
405C22—4882no—
406C22#1683999912yes—
407C22C2×D8×D6111yescharacteristic
408C4—82no—
409C4—81no—
410C4—81no—
411C4—1222no—
412C4—4881no—
413C4#1683999912yes—
414C4C2×#1451975211yescharacteristic
415C4C2×#1451980011yescharacteristic
416C2—22no—
417C2—9762no—
418C2—9761no—
419C2—2442no—
420C2C2×#1683999911yescharacteristic
421C2#2145895911yescharacteristic
422C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
hypercenter#2159D61
hypocenterC61#33612
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×#16840180C2semidirect
3. #21455491C2semidirect
4. #21455468C2semidirect
5. #21447933C2semidirect
6. #21447923C2semidirect
7. #21447916C2semidirect
8. #21447906C2semidirect
9. #21447524C2semidirect
10. C2×#16839562C2semidirect
11. C61×#2159C2semidirect
21. C2×#14519716C22semidirect
23. C2×C16×D61C22semidirect
26. #16840004C22semidirect
29. #16839999C22semidirect
32. #16839565C22semidirect
33. #16839565C22semidirect
36. #16839562C22semidirect
37. #16839562C22semidirect
41. C61×#367C22semidirect
42. C61×#366C22semidirect
84. M32×C61D4semidirect
213. #2159D61semidirect
246. #367C2×D61semidirect
247. #366C2×D61semidirect
254. C61#33612semidirect
288. M32D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×#168401802
#214554912
#214554682
#214479332
#214479232
#214479162
#214479062
#214475242
C2×#168395622
C61×#21592
#3361261