Finite Group Databasebeta

← #29722587

Subgroups — #29722587

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840037C211yesmaximal, characteristic, complemented
2#21455451C212yesmaximal
3C2×#16840020C211yesmaximal, characteristic, complemented
4C2×#16839811C211yesmaximal, characteristic, complemented
5#21447185C212yesmaximal, complemented
6#21447117C212yesmaximal, complemented
7#21446734C211yesmaximal, characteristic, complemented
8#21446599C211yesmaximal, characteristic, complemented
9#21446285C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C61×#2930C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C22×Dic244C2211yescharacteristic
14C22×#11758194C2212yes—
15C2×#14519756C2212yes—
16#16840037—24no—
17#16840020—22no—
18#16840020—22no—
19#16840018—22no—
20#16840006—22no—
21#16840001—22no—
22#16839989—22no—
23#16839811—24no—
24C2×#14519713C2212yes—
25C2×#14519713C2211yescharacteristic
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519709C2211yescharacteristic
29#16839770C2212yes—
30#16839763C2212yes—
31C2×#14519702C2212yes—
32C2×#14519695C2212yes—
33C2×#14519695C2212yes—
34#16839530C2212yes—
35#16839530C2212yes—
36C2×#131×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C61×#525C2211yescharacteristic
39C61×#436—22no—
40C61×#435—22no—
41C61×#424C2211yescharacteristic
42C61×#408C2211yescharacteristic
43C61×#407C2211yescharacteristic
44C61×#405C2211yescharacteristic
45C22×Dic122C2312yes—
46#14519807—24no—
47#14519807—42no—
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×Dic244—22no—
51C2×Dic244—22no—
52C2×#11758202—22no—
53C2×#11758200—22no—
54C2×#11758194—22no—
55C2×#11758194—24no—
56C2×#11758194—42no—
57C2×#11758194C2312yes—
58C2×#11758194C2312yes—
59C22×#92418—22no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—22no—
66C2×C22⋊Dic61—22no—
67#14519713—22no—
68#14519713—24no—
69#14519713D414yescomplemented
70#14519710—22no—
71#14519709—21no—
72#14519709—21no—
73#14519708—22no—
74#14519702—24no—
75#14519701—22no—
76#14519700—22no—
77#14519700—22no—
78#14519698—22no—
79#14519695—24no—
80#14519695—42no—
81C2×#41×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#131×C61—22no—
84#131×C61—22no—
85C2×M16×C61—22no—
86C2×M16×C61—21no—
87C2×M16×C61—22no—
88C2×M16×C61D412yes—
89C2×M16×C61D412yes—
90C2×M16×C61C2311yescharacteristic
91C2×M16×C61C2311yescharacteristic
92C22×C8×C61—21no—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95#123×C61—21no—
96C22⋊Q8×C61D412yes—
97#121×C61D412yes—
98#120×C61—21no—
99C4×D4×C61—22no—
100#117×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C22⋊C8×C61—22no—
103C22⋊C8×C61—22no—
104#97×C61—21no—
105C4×C8×C61—21no—
106C24×C61#4111yescharacteristic
107#41×C61—21no—
108#41×C61—21no—
109#41×C61—21no—
110#41×C61—21no—
111C2×Q8×C61C2×D411yescharacteristic
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61C2×D411yescharacteristic
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—22no—
125C22×Dic61—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132M16×C61—24no—
133M16×C61—24no—
134M16×C61—21no—
135M16×C61—21no—
136M16×C61—21no—
137M16×C61—21no—
138M16×C61—42no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—42no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D412yes—
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61#4112yes—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—21no—
153C22⋊C4×C61—22no—
154C42×C61—21no—
155C22⋊Dic61—42no—
156C22⋊Dic61—42no—
157#11758202—44no—
158#11758200—44no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161#11758196—42no—
162#11758196—42no—
163#11758194—22no—
164#11758194—22no—
165#11758194—22no—
166#11758194—44no—
167#11758194C2×D414yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—44no—
171Dic244—42no—
172Dic244—42no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—42no—
177C23×C61#12112yes—
178C23×C61C2×#4111yescharacteristic
179Q8×C61—21no—
180Q8×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—82no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61#12112yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—82no—
208Dic122—42no—
209Dic122—42no—
210Dic122—42no—
211Dic122—42no—
212C8×C61—22no—
213C8×C61—41no—
214C8×C61—41no—
215C8×C61#12112yes—
216C8×C61#12112yes—
217#92418—24no—
218#92418—42no—
219#24807—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—41no—
225C22×C61—42no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61C2×#12111yescharacteristic
229C22×C61#54811yescharacteristic
230C22×C61#55311yescharacteristic
231C4×C61—22no—
232C4×C61—41no—
233C4×C61—42no—
234C4×C61C2×#12111yescharacteristic
235C4×C61#53711yescharacteristic
236C4×C61D4211yescharacteristic
237C4×C61#54711yescharacteristic
238Dic61—82no—
239Dic61—82no—
240C2×#473—611no—
241#3056—612no—
242C2×#470—611no—
243#2930D6111yescharacteristic
244C2×#430—611no—
245#2064—612no—
246#2052—612no—
247#1979—611no—
248#1951—611no—
249#1901—612no—
250#1815—611no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—42no—
254C2×C61—81no—
255C2×C61#241611yescharacteristic
256C2×C61#306912yes—
257C2×#137—612no—
258C22×Q16—611no—
259C2×#131C2×D6111yescharacteristic
260C22×M16—612no—
261C22×M16C2×D6111yescharacteristic
262#525C2×D6111yescharacteristic
263C2×C22⋊Q8—612no—
264#473—1224no—
265#470—1222no—
266#470—1222no—
267#470—1222no—
268#468—1222no—
269#467—1222no—
270#465—1222no—
271#436—22no—
272#435—22no—
273#430—1224no—
274C2×#108—611no—
275C2×#108—612no—
276C2×#108—611no—
277#428—612no—
278C2×#106—611no—
279#424C2×D6111yescharacteristic
280#419—612no—
281#417—612no—
282C2×#103—612no—
283C2×#101—612no—
284C2×#101—612no—
285#408C2×D6111yescharacteristic
286#407C2×D6111yescharacteristic
287#405C2×D6111yescharacteristic
288#343—612no—
289#343—612no—
290C61#2480711yescharacteristic, complemented, Sylow
291C2×#41C22×D6111yescharacteristic
292C22×Q8—612no—
293C23×C4C22×D6111yescharacteristic
294#137—1224no—
295#137—2442no—
296C2×Q16—1222no—
297C2×Q16—1222no—
298C2×Q16—1222no—
299C2×Q16—1221no—
300C2×Q16—1221no—
301C2×SD16—1222no—
302#131—22no—
303#131—22no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—21no—
307C2×M16—612no—
308C2×M16—612no—
309C2×M16—1222no—
310C2×M16—1224no—
311C2×M16—2442no—
312C2×M16C22⋊D6112yes—
313C2×M16C22⋊D6112yes—
314C2×M16C22×D6111yescharacteristic
315C2×M16C22×D6111yescharacteristic
316C22×C8—21no—
317C22×C8—1222no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320#123—21no—
321C22⋊Q8—1222no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C22⋊Q8D24412yes—
326#121D24412yes—
327#120—21no—
328C4×D4—22no—
329#117C22×D6111yescharacteristic
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C22⋊C4—1222no—
333#108—614no—
334#108—1224no—
335#108—1222no—
336#107—1222no—
337#106—1221no—
338#106—1221no—
339#106—1222no—
340C4⋊C8—22no—
341#103—1222no—
342#103—1224no—
343#103—1222no—
344#103—1222no—
345#102—1222no—
346#101—1224no—
347#101—2442no—
348C22⋊C8—22no—
349C22⋊C8—22no—
350#97—21no—
351C4×C8—21no—
352C24#1175822211yescharacteristic
353#41—21no—
354#41—21no—
355#41—21no—
356#41—21no—
357C2×Q8—1222no—
358C2×Q8—1222no—
359C2×Q8—1222no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8—1222no—
363C2×Q8C2×D24411yescharacteristic
364C2×D4—21no—
365C2×D4—22no—
366C2×D4C2×D24411yescharacteristic
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4C2×D24412yes—
375C22×C4#1175822211yescharacteristic
376C22×C4C2×C22⋊D6112yes—
377C22×C4C23×D6111yescharacteristic
378Q16—2444no—
379Q16—2442no—
380Q16—2442no—
381SD16—2444no—
382M16—24no—
383M16—24no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—42no—
389M16—614no—
390M16—1222no—
391M16—1222no—
392M16—1222no—
393M16—2444no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—42no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—2444no—
402C2×C8D4×D6112yes—
403C2×C8#1175822412yes—
404C2×C8C2×C22⋊D6112yes—
405C2×C8C2×C22⋊D6111yescharacteristic
406C2×C8C2×C22⋊D6111yescharacteristic
407C4⋊C4—22no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2442no—
411C4⋊C4—2442no—
412C4⋊C4C2×D24412yes—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C42—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—42no—
423C23#1451975112yes—
424C23C2×#1175822211yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—2442no—
428Q8—2442no—
429Q8—2442no—
430Q8—2442no—
431D4—21no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—82no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—4882no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4#1451975112yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C22×D24411yescharacteristic
455C2×C4C2×#1175822411yescharacteristic
456C2×C4#1451982311yescharacteristic
457C2×C4#1451982411yescharacteristic
458C8—22no—
459C8—41no—
460C8—41no—
461C8—1224no—
462C8—2442no—
463C8#1451974912yes—
464C8#1451974912yes—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—41no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22C2×#1451975111yescharacteristic
474C22#1684042111yescharacteristic
475C22#1684045811yescharacteristic
476C4—22no—
477C4—41no—
478C4—42no—
479C4—4882no—
480C4—4882no—
481C4C2×#1451974911yescharacteristic
482C4#1684034011yescharacteristic
483C4D4×D24411yescharacteristic
484C4#1684041611yescharacteristic
485C2—21no—
486C2—21no—
487C2—42no—
488C2—81no—
489C2#2144925911yescharacteristic
490C2#2145555612yes—
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2930C2
socleC22×C61C2×#121
prime radicalC61×#525C22
hypercenter#2930D61
hypocenterC61#24807
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2930

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
3. C2×#16840020C2semidirect
4. C2×#16839811C2semidirect
5. #21447185C2semidirect
6. #21447117C2semidirect
7. #21446734C2semidirect
8. #21446599C2semidirect
9. #21446285C2semidirect
10. #21445878C2semidirect
69. #14519713D4semidirect
290. C61#24807semidirect

Maximal subgroups

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

subgroupindex
C2×#168400372
#214554512
C2×#168400202
C2×#168398112
#214471852
#214471172
#214467342
#214465992
#214462852
#214458782
C61×#29302
#2480761