Finite Group Databasebeta

← #29834598

Subgroups — #29834598

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840198C211yesmaximal, characteristic, complemented
2#21458921C211yesmaximal, characteristic, complemented
3#21455540C211yesmaximal, characteristic, complemented
4C2×#16840029C211yesmaximal, characteristic, complemented
5#21447948C214yesmaximal, complemented
6#21447938C214yesmaximal, complemented
7#21447613C211yesmaximal, characteristic, complemented
8#21447569C211yesmaximal, characteristic, complemented
9C2×C61×#361C211yesmaximal, characteristic, complemented
10#16840562—21no—
11C2×#14519803—21no—
12C2×#14519799C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840198—22no—
15#16840198—22no—
16C2×#14519721C2212yescomplemented
17C2×#14519718C2212yescomplemented
18#16840180C2214yescomplemented
19#16840178—22no—
20C2×#14519717C2211yescharacteristic
21M32×D61—22no—
22#16840038C2212yescomplemented
23#16840033C2212yescomplemented
24#16840029—22no—
25#16840029C2214yes—
26#16840026—22no—
27C2×#14519713C2211yescharacteristic
28#16839744—21no—
29C2×#14519694—21no—
30#16839591C2214yescomplemented
31#16839579C2212yescomplemented
32#16839576C2212yescomplemented
33C22×D8×C61C2211yescharacteristic
34C2×M32×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#361C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222D411yescharacteristic
39#14519810—22no—
40#14519810—41no—
41#14519805—22no—
42#14519805—41no—
43#14519803—24no—
44#14519803—42no—
45#14519799—21no—
46#14519799—21no—
47#14519799—21no—
48#14519799—21no—
49#14519799—42no—
50C2×Dic244C2311yescharacteristic
51C2×#11758190—21no—
52C2×#11758200—41no—
53C2×D488C2311yescharacteristic
54C2×#11758199—41no—
55#14519783—21no—
56#14519783—21no—
57#14519783D412yes—
58#14519783C2312yes—
59C2×#11758194—21no—
60C2×#11758194D411yescharacteristic
61C2×#11758189—21no—
62C2×C8×D61—21no—
63#14519721—24no—
64#14519718—24no—
65#14519717D412yes—
66#14519717D412yes—
67#14519716—24no—
68C2×#11758185C2312yes—
69C16×D61—24no—
70#14519713—21no—
71#14519713C2312yes—
72#14519695—22no—
73#14519694—24no—
74#14519693—22no—
75C22×D4×C61—21no—
76C2×D8×C61—21no—
77C2×D8×C61—21no—
78C2×D8×C61—21no—
79C2×D8×C61—21no—
80C2×D8×C61—21no—
81C2×D8×C61C2312yes—
82C2×M16×C61—21no—
83C22×C8×C61C2311yescharacteristic
84M32×C61—21no—
85M32×C61C2312yes—
86C2×C16×C61—21no—
87#108×C61—21no—
88#108×C61C2312yes—
89C24×C61—41no—
90C2×D4×C61—22no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—21no—
98C2×C22⋊D61—41no—
99#11758222—21no—
100#11758222—21no—
101#11758222—41no—
102#11758222—41no—
103#11758222—41no—
104#11758222—41no—
105#11758222D812yes—
106#11758222C2×D412yes—
107C2×D244—21no—
108C2×D244D811yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—41no—
111C2×Dic122—21no—
112C2×Dic122D811yescharacteristic
113D8×C61—22no—
114D8×C61—22no—
115D8×C61—41no—
116D8×C61—41no—
117M16×C61—22no—
118M16×C61—41no—
119C2×C8×C61—41no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61#4111yescharacteristic
124C2×C8×C61#4111yescharacteristic
125C2×C8×C61C2411yescharacteristic
126#11758200—44no—
127#11758199—44no—
128#11758194—21no—
129#11758194—22no—
130#11758194—41no—
131#11758194C2×D412yes—
132C2×#92418—21no—
133C2×#92418—41no—
134Dic244—21no—
135Dic244—21no—
136Dic244—42no—
137D488—21no—
138D488—21no—
139D488—42no—
140#11758190—42no—
141#11758190—42no—
142#11758190—41no—
143#11758190—41no—
144#11758189—22no—
145#11758189—22no—
146C8×D61—22no—
147C8×D61—22no—
148C16×C61—22no—
149#11758185C2×D414yes—
150C23×C61—42no—
151C23×C61—81no—
152C23×C61—81no—
153C23×C61—162no—
154C23×C61#13511yescharacteristic
155C22×D61—21no—
156C22×D61—41no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—81no—
160D4×C61—81no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C2×D811yescharacteristic
164C2×C4×C61C2×D811yescharacteristic
165C2×C4×C61#13511yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C22⋊D61—22no—
168C22⋊D61—22no—
169C22⋊D61—81no—
170C22⋊D61—81no—
171C2×Dic61—21no—
172C2×Dic61—41no—
173D244—21no—
174D244—41no—
175D244—41no—
176D244C2×D812yes—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—81no—
180C4×D61—81no—
181Dic122—21no—
182Dic122—41no—
183Dic122—41no—
184Dic122C2×D812yes—
185C8×C61—42no—
186C8×C61#12111yescharacteristic
187C8×C61#12111yescharacteristic
188C8×C61C22×D411yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—22no—
191#92418—42no—
192#33618—611nomaximal, Sylow
193C22×C61—41no—
194C22×C61—82no—
195C22×C61—162no—
196C22×C61—162no—
197C22×C61—162no—
198C22×C61#46611yescharacteristic
199C22×C61#46611yescharacteristic
200C22×C61C2×#13511yescharacteristic
201C2×D61—22no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—161no—
205C4×C61#46611yescharacteristic
206C4×C61#46611yescharacteristic
207C4×C61C2×#12111yescharacteristic
208C4×C61C22×D811yescharacteristic
209Dic61—22no—
210Dic61—81no—
211C2×#509—611no—
212#3407—611no—
213#3067—611no—
214C2×#472—611no—
215#2223—614no—
216#2221—614no—
217#2161—611no—
218#2148—611no—
219C2×#361D6111yescharacteristic, complemented
220C2×C61—21no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—162no—
224C2×C61C2×#46611yescharacteristic
225D61—42no—
226D61—161no—
227#575—1221no—
228C2×AGL(1,ℤ/8)—1221no—
229C2×#135—611no—
230C22×D8C2×D6111yescharacteristic
231C2×#131—611no—
232#509—1222no—
233#509—1222no—
234C2×SD32—612no—
235C2×D16—612no—
236#504—614no—
237#504—1222no—
238C2×M32—611no—
239C2×M32—1222no—
240C2×M32C2×D6111yescharacteristic
241#473—612no—
242#472—614no—
243#472—612no—
244#472—1222no—
245#471—1222no—
246C2×#108—611no—
247C2×#108C2×D6111yescharacteristic
248#413—1221no—
249C2×#100—1221no—
250#368—614no—
251#362—612no—
252#361—612no—
253#361C2×D6114yescomplemented
254C61#3361811yescharacteristic, complemented, Sylow
255C2×#41—611no—
256C2×#41—1221no—
257C22×D4—21no—
258#137—1222no—
259#137—2441no—
260AGL(1,ℤ/8)—1222no—
261AGL(1,ℤ/8)—1224no—
262AGL(1,ℤ/8)—2441no—
263AGL(1,ℤ/8)—2442no—
264#135—1221no—
265#135—1221no—
266#135—1221no—
267#135—1221no—
268#135—2442no—
269C2×Q16—611no—
270C2×SD16—1221no—
271C2×SD16—2441no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—611no—
278C2×D8—2441no—
279C2×D8C22×D6112yes—
280#131—612no—
281#131—612no—
282#131—1221no—
283#131—1221no—
284C2×M16—21no—
285C2×M16—611no—
286C2×M16—1221no—
287C2×M16—1221no—
288C22×C8—1221no—
289C22×C8C22×D6111yescharacteristic
290SD32—1224no—
291D16—1224no—
292M32—21no—
293M32—612no—
294M32—612no—
295M32—1224no—
296M32C22×D6112yes—
297C2×C16—21no—
298C2×C16—612no—
299C2×C16—1224no—
300#108—21no—
301#108—612no—
302#108—1221no—
303#108C22×D6112yes—
304#101—1222no—
305#100—1222no—
306#100—1224no—
307C24—41no—
308#41—612no—
309#41—612no—
310#41—1221no—
311#41—1221no—
312#41—2441no—
313#41—2441no—
314#41—2441no—
315#41—2441no—
316C2×Q8—611no—
317C2×Q8—1221no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—611no—
325C2×D4—1221no—
326C2×D4—1221no—
327C2×D4—2441no—
328C22×C4—1221no—
329C22×C4—2441no—
330C22×C4D4×D6111yescharacteristic
331Q16—1221no—
332Q16—1221no—
333Q16—2442no—
334SD16—2442no—
335SD16—2442no—
336SD16—2444no—
337SD16—2441no—
338SD16—2441no—
339D8—22no—
340D8—22no—
341D8—41no—
342D8—41no—
343D8—1221no—
344D8—1221no—
345D8—2442no—
346D8—2444no—
347M16—22no—
348M16—41no—
349M16—612no—
350M16—1221no—
351M16—1222no—
352M16—1222no—
353M16—1222no—
354M16—2441no—
355C2×C8—41no—
356C2×C8—1221no—
357C2×C8—1222no—
358C2×C8—1222no—
359C2×C8—2441no—
360C2×C8#1175822211yescharacteristic
361C2×C8#1175822211yescharacteristic
362C2×C8D4×D6111yescharacteristic
363C2×C8D4×D6111yescharacteristic
364C2×C8D4×D6111yescharacteristic
365C2×C8C23×D6111yescharacteristic
366C16—22no—
367C16—614no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—162no—
372C23—1221no—
373C23—2441no—
374C23#1451979811yescharacteristic
375Q8—612no—
376Q8—1221no—
377Q8—2441no—
378Q8—2441no—
379D4—42no—
380D4—42no—
381D4—81no—
382D4—81no—
383D4—612no—
384D4—1222no—
385D4—1222no—
386D4—1221no—
387D4—4881no—
388D4—4881no—
389D4—2441no—
390D4—2441no—
391C2×C4—1221no—
392C2×C4—1222no—
393C2×C4—1222no—
394C2×C4—4881no—
395C2×C4—4881no—
396C2×C4—2441no—
397C2×C4#1451975011yescharacteristic
398C2×C4#1451975011yescharacteristic
399C2×C4D8×D6111yescharacteristic
400C2×C4D8×D6111yescharacteristic
401C2×C4#1451979811yescharacteristic
402C2×C4C2×D4×D6111yescharacteristic
403C8—42no—
404C8—1222no—
405C8—2442no—
406C8#1451975011yescharacteristic
407C8#1451975011yescharacteristic
408C8C2×D4×D6111yescharacteristic
409C8C2×#1175822211yescharacteristic
410C22—41no—
411C22—82no—
412C22—162no—
413C22—162no—
414C22—162no—
415C22—1222no—
416C22—9761no—
417C22—4881no—
418C22—4881no—
419C22#1683999711yescharacteristic
420C22#1683999711yescharacteristic
421C22C2×#1451979811yescharacteristic
422C4—1222no—
423C4—4881no—
424C4#1683999711yescharacteristic
425C4#1683999711yescharacteristic
426C4C2×#1451975011yescharacteristic
427C4C2×D8×D6111yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—162no—
432C2—9761no—
433C2—2442no—
434C2C2×#1683999711yescharacteristic
subgroupG/H
centerC2C2×#16839997
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#361C2
socleC2×C61C2×#466
hypercenterC2×#361D61
hypocenterC61#33618
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. #21458921C2semidirect
3. #21455540C2semidirect
4. C2×#16840029C2semidirect
5. #21447948C2semidirect
6. #21447938C2semidirect
7. #21447613C2semidirect
8. #21447569C2semidirect
9. C2×C61×#361C2semidirect
16. C2×#14519721C22semidirect
17. C2×#14519718C22semidirect
18. #16840180C22semidirect
22. #16840038C22semidirect
23. #16840033C22semidirect
30. #16839591C22semidirect
31. #16839579C22semidirect
32. #16839576C22semidirect
36. C61×#361C22semidirect
219. C2×#361D61semidirect
253. #361C2×D61semidirect
254. C61#33618semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
#214589212
#214555402
C2×#168400292
#214479482
#214479382
#214476132
#214475692
C2×C61×#3612
#3361861