Finite Group Databasebeta

← #29834604

Subgroups — #29834604

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458921C211yesmaximal, characteristic, complemented
2#21455535C211yesmaximal, characteristic
3C2×#16840034C211yesmaximal, characteristic, complemented
4#21447946C214yesmaximal, complemented
5#21447941C214yesmaximal, complemented
6#21447614C211yesmaximal, characteristic, complemented
7#21447568C211yesmaximal, characteristic, complemented
8C2×#16839578C211yesmaximal, characteristic, complemented
9C2×C61×#510C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519808—21no—
12C2×#14519783C2211yescharacteristic
13#16840180—22no—
14#16840178C2214yes—
15C2×#14519717C2211yescharacteristic
16M32×D61—22no—
17#16840034—22no—
18#16840034C2212yes—
19#16840034C2214yes—
20#16840029C2212yes—
21#16840022—22no—
22C2×#14519713C2211yescharacteristic
23C2×#14519713C2211yescharacteristic
24#16839746—21no—
25C2×#14519696—21no—
26#16839592C2214yescomplemented
27#16839578C2212yescomplemented
28#16839578C2214yescomplemented
29#16839575C2212yescomplemented
30C2×#135×C61C2211yescharacteristic
31C22×Q16×C61C2211yescharacteristic
32C61×#510—22no—
33C61×#510—22no—
34C2×Q32×C61C2212yescomplemented
35C2×SD32×C61C2212yescomplemented
36C2×M32×C61C2211yescharacteristic
37C2×#11758222D411yescharacteristic
38#14519808—22no—
39#14519808—24no—
40#14519808—42no—
41#14519807—41no—
42#14519803—22no—
43#14519802—41no—
44#14519797—42no—
45C2×#11758202—41no—
46C2×#11758201—41no—
47#14519783—21no—
48#14519783—21no—
49#14519783D412yes—
50#14519783C2312yes—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194D411yescharacteristic
54C2×#11758189—21no—
55C2×C8×D61—21no—
56#14519717—21no—
57#14519717C2312yes—
58#14519716—24no—
59C2×#11758185—21no—
60C16×D61—24no—
61#14519713—21no—
62#14519713—21no—
63#14519713C2312yes—
64#14519713C2312yes—
65#14519696—24no—
66#14519696—22no—
67#14519694—22no—
68C2×#41×C61—21no—
69C22×Q8×C61—21no—
70#135×C61—21no—
71#135×C61—21no—
72#135×C61—21no—
73#135×C61—21no—
74C2×Q16×C61—21no—
75C2×Q16×C61—21no—
76C2×Q16×C61—21no—
77C2×Q16×C61—21no—
78C2×Q16×C61—21no—
79C2×Q16×C61C2312yes—
80C2×Q16×C61C2311yescharacteristic
81C2×SD16×C61—21no—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84Q32×C61—24no—
85SD32×C61—24no—
86M32×C61D412yes—
87M32×C61D412yes—
88C2×C16×C61C2312yes—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93C2×Q8×C61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—41no—
96C2×Q8×C61—41no—
97C2×Q8×C61—41no—
98C2×Q8×C61—41no—
99C2×Q8×C61—41no—
100C2×D4×C61—21no—
101C2×D4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—21no—
106#11758222—21no—
107#11758222—21no—
108#11758222D812yes—
109#11758222C2×D412yes—
110C2×D244D811yescharacteristic
111C2×C4×D61—21no—
112C2×Dic122D811yescharacteristic
113Q16×C61—22no—
114Q16×C61—21no—
115Q16×C61—22no—
116Q16×C61—21no—
117Q16×C61—41no—
118Q16×C61—41no—
119SD16×C61—41no—
120SD16×C61—41no—
121D8×C61—21no—
122D8×C61—21no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61#4111yescharacteristic
127C2×C8×C61#4111yescharacteristic
128C2×C8×C61C2411yescharacteristic
129#11758202—44no—
130#11758202—42no—
131#11758201—44no—
132#11758201—42no—
133#11758200—42no—
134#11758199—42no—
135#11758194—21no—
136#11758194—22no—
137#11758194—22no—
138#11758194—41no—
139#11758194—41no—
140#11758194C2×D412yes—
141C2×#92418—21no—
142C2×#92418—41no—
143C2×#92418—41no—
144#11758189—22no—
145#11758189—22no—
146C8×D61—22no—
147C8×D61—22no—
148C16×C61C2×D414yes—
149#11758185—22no—
150C23×C61—41no—
151C23×C61#13511yescharacteristic
152C22×D61—21no—
153Q8×C61—41no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61—41no—
157Q8×C61—81no—
158Q8×C61—81no—
159D4×C61—41no—
160D4×C61—41no—
161D4×C61—81no—
162D4×C61—81no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×D811yescharacteristic
172C2×C4×C61C2×D811yescharacteristic
173C2×C4×C61#13511yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C22⋊D61—22no—
176C22⋊D61—22no—
177C2×Dic61—21no—
178D244—21no—
179D244C2×D812yes—
180C4×D61—22no—
181C4×D61—22no—
182Dic122—21no—
183Dic122C2×D812yes—
184C8×C61#12111yescharacteristic
185C8×C61#12111yescharacteristic
186C8×C61C22×D411yescharacteristic
187C8×C61C2×#4111yescharacteristic
188#92418—22no—
189#92418—42no—
190#92418—42no—
191#33619—611nomaximal, Sylow
192C22×C61—41no—
193C22×C61—81no—
194C22×C61—161no—
195C22×C61#46611yescharacteristic
196C22×C61#46611yescharacteristic
197C22×C61C2×#13511yescharacteristic
198C2×D61—22no—
199C2×D61—81no—
200C4×C61—81no—
201C4×C61—82no—
202C4×C61#46611yescharacteristic
203C4×C61#46611yescharacteristic
204C4×C61C2×#12111yescharacteristic
205C4×C61C22×D811yescharacteristic
206Dic61—22no—
207C2×#510D6111yescharacteristic, complemented
208#3407—611no—
209#3067—611no—
210C2×#473—611no—
211#2223—614no—
212#2222—614no—
213#2161—611no—
214#2148—611no—
215C2×#362—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61C2×#46611yescharacteristic
221D61—42no—
222#575—1221no—
223C2×#137—1221no—
224C2×#135C2×D6111yescharacteristic
225C22×Q16C2×D6111yescharacteristic
226C2×#131—611no—
227#510—22no—
228#510—22no—
229C2×Q32C2×D6112yescomplemented
230C2×SD32C2×D6112yescomplemented
231#504—614no—
232#504—1222no—
233C2×M32—611no—
234C2×M32—1222no—
235C2×M32C2×D6111yescharacteristic
236#473—612no—
237#473—614no—
238#473—1222no—
239#472—612no—
240#471—1222no—
241C2×#108—611no—
242C2×#108—611no—
243#413—1221no—
244C2×#101—1221no—
245#368—614no—
246#362—614no—
247#362—612no—
248#361—612no—
249C61#3361911yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41—611no—
252C22×Q8—21no—
253#137—1224no—
254#137—1222no—
255#137—2442no—
256#137—2441no—
257AGL(1,ℤ/8)—1222no—
258AGL(1,ℤ/8)—2441no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—2442no—
264C2×Q16—21no—
265C2×Q16—21no—
266C2×Q16—21no—
267C2×Q16—21no—
268C2×Q16—21no—
269C2×Q16—2441no—
270C2×Q16C22×D6112yes—
271C2×Q16C22×D6111yescharacteristic
272C2×SD16—21no—
273C2×SD16—2441no—
274C2×D8C22×D6111yescharacteristic
275#131—612no—
276#131—612no—
277#131—1221no—
278#131—1221no—
279C2×M16—611no—
280C2×M16—1221no—
281C2×M16—1221no—
282C2×M16—1221no—
283C22×C8—1221no—
284C22×C8C22×D6111yescharacteristic
285Q32—24no—
286SD32—24no—
287M32—612no—
288M32—1221no—
289M32—1224no—
290M32C22⋊D6112yes—
291M32C22⋊D6112yes—
292C2×C16—1221no—
293C2×C16—1224no—
294C2×C16C22×D6112yes—
295#108—612no—
296#108—612no—
297#108—1221no—
298#108—1221no—
299#101—1224no—
300#101—1222no—
301#100—1222no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—612no—
307#41—612no—
308#41—1221no—
309#41—1221no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—611no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—611no—
321C2×D4—1221no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—1221no—
325C22×C4D4×D6111yescharacteristic
326Q16—22no—
327Q16—21no—
328Q16—22no—
329Q16—21no—
330Q16—41no—
331Q16—41no—
332Q16—2444no—
333Q16—2442no—
334SD16—41no—
335SD16—41no—
336SD16—2442no—
337SD16—2444no—
338SD16—2442no—
339D8—21no—
340D8—21no—
341D8—2442no—
342M16—612no—
343M16—1221no—
344M16—1222no—
345M16—1222no—
346M16—1222no—
347M16—1222no—
348M16—2441no—
349M16—2441no—
350C2×C8—1221no—
351C2×C8—1222no—
352C2×C8—1222no—
353C2×C8—2441no—
354C2×C8—2441no—
355C2×C8D4×D6111yescharacteristic
356C2×C8#1175822411yescharacteristic
357C2×C8#1175822411yescharacteristic
358C2×C8C2×C22⋊D6111yescharacteristic
359C2×C8C2×C22⋊D6111yescharacteristic
360C2×C8C23×D6111yescharacteristic
361C16—1222no—
362C16C2×C22⋊D6114yes—
363C23—41no—
364C23—1221no—
365C23#1451979611yescharacteristic
366Q8—41no—
367Q8—42no—
368Q8—42no—
369Q8—41no—
370Q8—81no—
371Q8—81no—
372Q8—612no—
373Q8—1221no—
374D4—41no—
375D4—41no—
376D4—81no—
377D4—81no—
378D4—612no—
379D4—1222no—
380D4—1222no—
381D4—1221no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4#1451974811yescharacteristic
392C2×C4#1451974811yescharacteristic
393C2×C4D8×D6111yescharacteristic
394C2×C4D8×D6111yescharacteristic
395C2×C4#1451979611yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C8—1222no—
398C8—2442no—
399C8—2442no—
400C8#1451974811yescharacteristic
401C8#1451974811yescharacteristic
402C8C22×C22⋊D6111yescharacteristic
403C8C2×#1175822411yescharacteristic
404C22—41no—
405C22—81no—
406C22—161no—
407C22—1222no—
408C22—4881no—
409C22#1683999511yescharacteristic
410C22#1683999511yescharacteristic
411C22C2×#1451979611yescharacteristic
412C4—81no—
413C4—82no—
414C4—1222no—
415C4#1683999511yescharacteristic
416C4#1683999511yescharacteristic
417C4C2×#1451974811yescharacteristic
418C4C2×D8×D6111yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—161no—
423C2—2442no—
424C2C2×#1683999511yescharacteristic
subgroupG/H
centerC2C2×#16839995
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC2×C61C2×#466
prime radical#21455535C2
hypercenterC2×#510D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. #21458921C2semidirect
3. C2×#16840034C2semidirect
4. #21447946C2semidirect
5. #21447941C2semidirect
6. #21447614C2semidirect
7. #21447568C2semidirect
8. C2×#16839578C2semidirect
9. C2×C61×#510C2semidirect
26. #16839592C22semidirect
27. #16839578C22semidirect
28. #16839578C22semidirect
29. #16839575C22semidirect
34. C2×Q32×C61C22semidirect
35. C2×SD32×C61C22semidirect
207. C2×#510D61semidirect
229. C2×Q32C2×D61semidirect
230. C2×SD32C2×D61semidirect
249. C61#33619semidirect

Maximal subgroups

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

subgroupindex
#214589212
#214555352
C2×#168400342
#214479462
#214479412
#214476142
#214475682
C2×#168395782
C2×C61×#5102
#3361961