Finite Group Databasebeta

← #29708565

Subgroups — #29708565

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519712C211yesmaximal, characteristic, complemented
2#21454408C211yesmaximal, characteristic, complemented
3C2×#16839790C211yesmaximal, characteristic, complemented
4#21454339C211yesmaximal, characteristic
5C2×#16839756C211yesmaximal, characteristic
6C2×#16839744C211yesmaximal, characteristic
7#21447187C214yesmaximal, complemented
8#21447180C214yesmaximal, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519712—24no—
15C2×#14519712C2214yes—
16#16839805—22no—
17#16839802C2214yes—
18#16839798—22no—
19#16839790—22no—
20#16839790C2214yes—
21C2×#14519706—22no—
22#16839772C2212yes—
23#16839768C2212yes—
24#16839764—22no—
25#16839759—22no—
26#16839756—22no—
27#16839756—22no—
28C2×#14519700C2212yes—
29C2×#14519697C2212yes—
30#16839744—24no—
31C2×#14519695C2212yes—
32C2×#14519693C2212yes—
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#343C2214yes—
37C2×#11758222—22no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222Q814yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49C2×#11758194—21no—
50C2×#11758194—22no—
51C2×#11758194D414yes—
52C2×#11758194C2312yes—
53C22×#92418—22no—
54C22×#92418—21no—
55C22×#92418—22no—
56C22×#92418D412yes—
57C22×#92418D412yes—
58C22×#92418C2312yes—
59#14519736—21no—
60#14519736—21no—
61#14519736D412yes—
62#14519736C2312yes—
63C2×C4⋊Dic61D412yes—
64C2×C22⋊Dic61—21no—
65C2×C4×Dic61—21no—
66#14519712—28no—
67#14519710—24no—
68#14519708—24no—
69#14519706—28no—
70#14519702—24no—
71#14519700—24no—
72#14519699—24no—
73#14519697—24no—
74#14519695—24no—
75#14519693—24no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—22no—
80C2×M16×C61—22no—
81C2×M16×C61—22no—
82C2×M16×C61—22no—
83C2×M16×C61C2312yes—
84C2×M16×C61C2312yes—
85C22×C8×C61—21no—
86C22×C8×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90#117×C61—21no—
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—21no—
93C2×C42×C61—21no—
94C24×C61#4111yescharacteristic
95C23×D61—21no—
96C22×C4×C61—22no—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×Q812yes—
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4112yes—
105C22×C4×C61#4112yes—
106C22×C4×C61C2411yescharacteristic
107C2×C22⋊D61—24no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—41no—
110C2×C22⋊D61—42no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113#11758222—24no—
114#11758222—24no—
115#11758222—24no—
116#11758222—24no—
117#11758222—42no—
118#11758222—42no—
119#11758222—42no—
120#11758222—42no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244C2×Q812yes—
127C2×C4×D61—24no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61—42no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122C2×Q812yes—
137M16×C61—22no—
138M16×C61—22no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61—42no—
142M16×C61—42no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—42no—
146C2×C8×C61—42no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C42×C61—41no—
152C42×C61—41no—
153C22⋊Dic61—22no—
154C22⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61C2×D414yes—
157C4×Dic61—22no—
158C4×Dic61—22no—
159#11758194—24no—
160#11758194—22no—
161#11758194C2×D414yes—
162C2×#92418—24no—
163C2×#92418—24no—
164C2×#92418—24no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418C2×D414yes—
169C23×C61—42no—
170C23×C61#12111yescharacteristic
171C23×C61#12111yescharacteristic
172C23×C61C22⋊Q812yes—
173C23×C61#12612yes—
174C23×C61C2×#4111yescharacteristic
175C22×D61—22no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—82no—
179C2×C4×C61—42no—
180C2×C4×C61—82no—
181C2×C4×C61#12112yes—
182C2×C4×C61#12112yes—
183C2×C4×C61#12112yes—
184C2×C4×C61#12112yes—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61C22⋊Q812yes—
188C2×C4×C61C22⋊Q812yes—
189C2×C4×C61C22⋊Q812yes—
190C2×C4×C61C22⋊Q812yes—
191C2×C4×C61C22⋊Q812yes—
192C2×C4×C61#12612yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×Q811yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C22⋊D61—44no—
200C22⋊D61—44no—
201C2×Dic61—24no—
202C2×Dic61—22no—
203C2×Dic61—44no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210C4×D61—44no—
211C4×D61—44no—
212Dic122—22no—
213Dic122—22no—
214Dic122—41no—
215Dic122—41no—
216C8×C61—42no—
217C8×C61—42no—
218#92418—24no—
219#92418—24no—
220#23601—611nomaximal, Sylow
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61#39612yes—
227C22×C61#39612yes—
228C22×C61C2×#12111yescharacteristic
229C22×C61C2×C22⋊Q811yescharacteristic
230C22×C61C2×#12611yescharacteristic
231C2×D61—42no—
232C2×D61—82no—
233C2×D61—82no—
234C2×D61—82no—
235C4×C61—82no—
236C4×C61#39612yes—
237C4×C61#39612yes—
238C4×C61C2×#12111yescharacteristic
239C4×C61C2×#12111yescharacteristic
240C4×C61C2×C22⋊Q811yescharacteristic
241C4×C61C2×C22⋊Q811yescharacteristic
242Dic61—42no—
243Dic61—44no—
244C22×#108—611no—
245#2911—611no—
246C2×#424—611no—
247#2896—611no—
248C2×#416—611no—
249C2×#413—611no—
250#2064—614no—
251#2063—614no—
252#1742D6111yescharacteristic, complemented
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—22no—
256C2×C61—22no—
257C2×C61C2×#39611yescharacteristic
258C2×C61#306712yes—
259D61—82no—
260C22×#41—611no—
261C22×M16—611no—
262C22×M16C2×D6111yescharacteristic
263C22×M16C2×D6111yescharacteristic
264C23×C8—611no—
265C2×#117—611no—
266C2×#117C2×D6111yescharacteristic
267C2×#108—614no—
268C2×#108—1224no—
269#428—614no—
270#428—1222no—
271#427—1222no—
272#424—614no—
273#424—1222no—
274C2×C4⋊C8—1222no—
275#419—612no—
276#418—612no—
277#417—1222no—
278#416—1222no—
279#416—1222no—
280#416—1222no—
281C2×#103—612no—
282C2×#102—612no—
283#413—1224no—
284C2×#101—612no—
285C2×#100—612no—
286#343C2×D6114yes—
287C61#2360111yescharacteristic, complemented, Sylow
288C2×#41—614no—
289C2×#41—1222no—
290C2×#41—1222no—
291C2×#41—1222no—
292C2×#41—1221no—
293C2×#41—1221no—
294C2×#41—1221no—
295C2×#41—1221no—
296C22×Q8—611no—
297C22×D4—611no—
298C22×D4—1221no—
299C23×C4—1221no—
300C23×C4C22×D6111yescharacteristic
301C2×M16—21no—
302C2×M16—21no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—22no—
307C2×M16—614no—
308C2×M16—612no—
309C2×M16—1222no—
310C2×M16—1221no—
311C2×M16C22×D6112yes—
312C2×M16C22×D6112yes—
313C22×C8—21no—
314C22×C8—21no—
315C22×C8—612no—
316C22×C8—612no—
317C22×C8—612no—
318C22×C8—1222no—
319C22×C8—1222no—
320C22×C8—1221no—
321#117—21no—
322#117—21no—
323#117—21no—
324#117—21no—
325#117—612no—
326#117—612no—
327#117—1221no—
328#117—1221no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4C22×D6112yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1221no—
333C2×C42—21no—
334C2×C42—1221no—
335#108—1228no—
336#107—1224no—
337#106—1224no—
338C4⋊C8—1228no—
339#103—1224no—
340#103—1224no—
341#102—1224no—
342#102—1224no—
343#101—1224no—
344#100—1224no—
345C24—1221no—
346C24#1175822211yescharacteristic
347#41—1224no—
348#41—1224no—
349#41—1224no—
350#41—1224no—
351#41—2442no—
352#41—2442no—
353#41—2442no—
354#41—2442no—
355C2×Q8—612no—
356C2×Q8—1221no—
357C2×Q8—1221no—
358C2×Q8—1221no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×D4—612no—
362C2×D4—1224no—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—2441no—
369C2×D4—2441no—
370C2×D4—2442no—
371C22×C4—22no—
372C22×C4—1221no—
373C22×C4—1222no—
374C22×C4—1224no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2442no—
378C22×C4#1175822212yes—
379C22×C4#1175822212yes—
380C22×C4#1175822211yescharacteristic
381C22×C4D4×D6112yes—
382C22×C4D4×D6111yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385C22×C4D4×D6111yescharacteristic
386C22×C4Q8×D6112yes—
387C22×C4C23×D6111yescharacteristic
388M16—22no—
389M16—22no—
390M16—22no—
391M16—22no—
392M16—42no—
393M16—42no—
394M16—614no—
395M16—1222no—
396M16—1224no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—614no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1224no—
406C2×C8—1224no—
407C2×C8—1224no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—614no—
411C4⋊C4—1222no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—41no—
417C42—41no—
418C42—1222no—
419C42—1222no—
420C23—42no—
421C23—1222no—
422C23—4882no—
423C23—2441no—
424C23—2441no—
425C23#1451975011yescharacteristic
426C23#1451975011yescharacteristic
427C23#1451975512yes—
428C23#1451977112yes—
429C23C2×#1175822211yescharacteristic
430Q8—1222no—
431Q8—1222no—
432Q8—2441no—
433Q8—2441no—
434D4—1222no—
435D4—1222no—
436D4—2444no—
437D4—2444no—
438D4—2441no—
439D4—2441no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—1224no—
443C2×C4—1222no—
444C2×C4—2444no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2444no—
448C2×C4—2444no—
449C2×C4#1451975011yescharacteristic
450C2×C4#1451975012yes—
451C2×C4#1451975012yes—
452C2×C4#1451975012yes—
453C2×C4#1451975012yes—
454C2×C4#1451975011yescharacteristic
455C2×C4#1451975512yes—
456C2×C4#1451975512yes—
457C2×C4#1451975512yes—
458C2×C4#1451975512yes—
459C2×C4#1451975512yes—
460C2×C4#1451977112yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4C2×Q8×D6111yescharacteristic
465C2×C4C2×#1175822211yescharacteristic
466C2×C4C2×#1175822211yescharacteristic
467C8—42no—
468C8—42no—
469C8—1224no—
470C8—1224no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—4882no—
477C22—4882no—
478C22—4882no—
479C22—2442no—
480C22#1683968612yes—
481C22#1683968612yes—
482C22C2×#1451975011yescharacteristic
483C22C2×#1451975511yescharacteristic
484C22C2×#1451977111yescharacteristic
485C4—82no—
486C4—2442no—
487C4—2444no—
488C4#1683968612yes—
489C4#1683968612yes—
490C4C2×#1451975011yescharacteristic
491C4C2×#1451975011yescharacteristic
492C4C2×#1451975511yescharacteristic
493C4C2×#1451975511yescharacteristic
494C2—21no—
495C2—21no—
496C2—22no—
497C2—22no—
498C2—4882no—
499C2C2×#1683968611yescharacteristic
500C2#2145553812yes—
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
hypercenter#1742D61
hypocenterC61#23601
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
1. C22×#14519712C2semidirect
2. #21454408C2semidirect
3. C2×#16839790C2semidirect
7. #21447187C2semidirect
8. #21447180C2semidirect
9. C61×#1742C2semidirect
252. #1742D61semidirect
287. C61#23601semidirect

Maximal subgroups

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

subgroupindex
C22×#145197122
#214544082
C2×#168397902
#214543392
C2×#168397562
C2×#168397442
#214471872
#214471802
C61×#17422
#2360161