Finite Group Databasebeta

← #29682558

Subgroups — #29682558

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454408C211yesmaximal, characteristic, complemented
2#21454339C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic
4C2×#16839744C211yesmaximal, characteristic
5C2×#16839720C211yesmaximal, characteristic, complemented
6#21446479C214yesmaximal, complemented
7#21446472C214yesmaximal, complemented
8C2×#16839526C211yesmaximal, characteristic, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839805—22no—
15#16839802C2214yes—
16#16839798—22no—
17#16839772C2212yes—
18#16839768C2212yes—
19#16839764—22no—
20#16839759—22no—
21#16839756—21no—
22#16839756—21no—
23#16839756—22no—
24C2×#14519700C2212yes—
25C2×#14519697C2212yes—
26#16839744C418yescomplemented
27C2×#14519695C2212yes—
28C2×#14519693C2212yes—
29#16839720—22no—
30#16839720C2214yes—
31C2×#14519687—21no—
32C2×C8×Dic61—21no—
33#16839526C2214yes—
34C22×M16×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#343C2214yes—
38C2×#11758222—22no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C2×#11758194—21no—
51C2×#11758194—22no—
52C2×#11758194C2×C414yes—
53C2×#11758194C2312yes—
54C22×#92418—21no—
55C22×#92418—22no—
56C22×#92418—22no—
57C22×#92418D411yescharacteristic
58C22×#92418D411yescharacteristic
59C22×#92418D412yes—
60C22×#92418C2312yes—
61#14519736—21no—
62#14519736—21no—
63#14519736D412yes—
64#14519736C2312yes—
65C2×C4⋊Dic61D412yes—
66C2×C22⋊Dic61—21no—
67C2×C4×Dic61—21no—
68#14519710—24no—
69#14519708—24no—
70#14519702—24no—
71#14519700—22no—
72#14519700—22no—
73#14519699—24no—
74#14519697—22no—
75#14519697—22no—
76#14519695C2×C418yescomplemented
77#14519693C2×C418yescomplemented
78#14519687—24no—
79C8×Dic61—24no—
80C23×C4×C61C2311yescharacteristic
81C2×M16×C61—21no—
82C2×M16×C61—22no—
83C2×M16×C61—21no—
84C2×M16×C61—22no—
85C2×M16×C61—22no—
86C2×M16×C61C2×C414yes—
87C2×M16×C61C2312yes—
88C2×M16×C61C2312yes—
89C22×C8×C61—21no—
90C22×C8×C61—21no—
91#117×C61—21no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—21no—
98C24×C61#4111yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61C22×C412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4112yes—
108C22×C4×C61#4112yes—
109C22×C4×C61#4112yes—
110C22×C4×C61C2411yescharacteristic
111C2×C22⋊D61—24no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C2×C22⋊D61—42no—
115C22×Dic61—22no—
116C22×Dic61—21no—
117#11758222—24no—
118#11758222—24no—
119#11758222—24no—
120#11758222—24no—
121#11758222—42no—
122#11758222—42no—
123#11758222—42no—
124#11758222—42no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C22×C412yes—
131C2×C4×D61—24no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—42no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122C22×C412yes—
141M16×C61—24no—
142M16×C61—22no—
143M16×C61—22no—
144M16×C61—22no—
145M16×C61—42no—
146M16×C61C22×C414yes—
147C2×C8×C61—24no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—42no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155C42×C61—41no—
156C42×C61—41no—
157C22⋊Dic61—22no—
158C22⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D414yes—
161C4×Dic61—22no—
162C4×Dic61—22no—
163#11758194—24no—
164#11758194—22no—
165#11758194C22×C414yes—
166C2×#92418—24no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—24no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418C2×D412yes—
174C2×#92418C2×D412yes—
175C23×C61—42no—
176C23×C61#11712yes—
177C23×C61#12111yescharacteristic
178C23×C61#12111yescharacteristic
179C23×C61#12612yes—
180C23×C61C2×#4111yescharacteristic
181C22×D61—22no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—82no—
185C2×C4×C61—42no—
186C2×C4×C61—82no—
187C2×C4×C61#11712yes—
188C2×C4×C61C4×D412yes—
189C2×C4×C61C4×D412yes—
190C2×C4×C61C4×D412yes—
191C2×C4×C61C4×D412yes—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12312yes—
195C2×C4×C61#12312yes—
196C2×C4×C61#12412yes—
197C2×C4×C61#12412yes—
198C2×C4×C61#12612yes—
199C2×C4×C61C23×C411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C22⋊D61—44no—
206C22⋊D61—44no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—44no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—44no—
217C4×D61—44no—
218Dic122—22no—
219Dic122—22no—
220Dic122—41no—
221Dic122—41no—
222C8×C61—24no—
223C8×C61—42no—
224#92418—24no—
225#92418—22no—
226#92418—22no—
227#21075—611nomaximal, Sylow
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61#38812yes—
234C22×C61#38812yes—
235C22×C61C2×#11711yescharacteristic
236C22×C61C2×#12111yescharacteristic
237C22×C61C2×#12611yescharacteristic
238C2×D61—42no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C4×C61—82no—
243C4×C61#38812yes—
244C4×C61#38812yes—
245C4×C61C2×C4×D411yescharacteristic
246C4×C61C2×C4×D411yescharacteristic
247C4×C61C2×#12311yescharacteristic
248C4×C61C2×#12411yescharacteristic
249Dic61—42no—
250Dic61—44no—
251#2911—611no—
252#2896—611no—
253C2×#416—611no—
254C2×#413—611no—
255C2×#405—611no—
256#1931—614no—
257#1930—614no—
258#1742D6111yescharacteristic, complemented
259C2×#341—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—22no—
263C2×C61—22no—
264C2×C61C2×#38811yescharacteristic
265C2×C61#306712yes—
266D61—82no—
267C22×#41—611no—
268C22×M16—611no—
269C22×M16C2×D6111yescharacteristic
270C22×M16C2×D6111yescharacteristic
271C23×C8—611no—
272C2×#117—611no—
273C2×#117C2×D6111yescharacteristic
274#428—614no—
275#428—1222no—
276#427—1222no—
277#419—612no—
278#418—612no—
279#417—1222no—
280#416—1221no—
281#416—1221no—
282#416—1222no—
283#416—1222no—
284C2×#103—612no—
285C2×#102—612no—
286#413—618no—
287C2×#101—612no—
288C2×#100—612no—
289#405—614no—
290#405—1222no—
291C2×#97—1221no—
292C2×C4×C8—1221no—
293#343C2×D6114yes—
294#341—614no—
295C61#2107511yescharacteristic, complemented, Sylow
296C2×#41—614no—
297C2×#41—1222no—
298C2×#41—1221no—
299C2×#41—1221no—
300C2×#41—1221no—
301C2×#41—1221no—
302C2×#41—1222no—
303C2×#41—1222no—
304C22×Q8—611no—
305C22×D4—611no—
306C22×D4—1221no—
307C23×C4—1221no—
308C23×C4C22×D6111yescharacteristic
309C2×M16—21no—
310C2×M16—22no—
311C2×M16—21no—
312C2×M16—22no—
313C2×M16—22no—
314C2×M16—614no—
315C2×M16—612no—
316C2×M16—1221no—
317C2×M16—1222no—
318C2×M16C4×D6114yes—
319C2×M16C22×D6112yes—
320C2×M16C22×D6112yes—
321C22×C8—21no—
322C22×C8—21no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8—612no—
326C22×C8—612no—
327C22×C8—1222no—
328C22×C8—1221no—
329C22×C8—1222no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—612no—
335#117—612no—
336#117—1221no—
337#117—1221no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4C22×D6112yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1221no—
342C2×C42—21no—
343C2×C42—1221no—
344#107—1224no—
345#106—1224no—
346#103—1222no—
347#103—1222no—
348#103—1224no—
349#102—1222no—
350#102—1222no—
351#102—1224no—
352#101—618no—
353#100—618no—
354#97—1224no—
355C4×C8—1224no—
356C24—1221no—
357C24#1175822211yescharacteristic
358#41—1224no—
359#41—1224no—
360#41—1224no—
361#41—1224no—
362#41—2442no—
363#41—2442no—
364#41—2442no—
365#41—2442no—
366C2×Q8—612no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1221no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—2441no—
380C2×D4—2441no—
381C2×D4—2442no—
382C22×C4—22no—
383C22×C4—1221no—
384C22×C4—1222no—
385C22×C4—1224no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2442no—
389C22×C4C2×C4×D6112yes—
390C22×C4#1175822212yes—
391C22×C4#1175822211yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4D4×D6111yescharacteristic
396C22×C4#1175822412yes—
397C22×C4#1175822612yes—
398C22×C4C23×D6111yescharacteristic
399M16—22no—
400M16—24no—
401M16—22no—
402M16—22no—
403M16—42no—
404M16—614no—
405M16—1222no—
406M16—1224no—
407M16C2×C4×D6114yes—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—42no—
412C2×C8—612no—
413C2×C8—612no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1224no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1224no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—614no—
424C4⋊C4—1222no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—1222no—
428C22⋊C4—1222no—
429C42—41no—
430C42—41no—
431C42—1222no—
432C42—1222no—
433C23—42no—
434C23—1222no—
435C23—4882no—
436C23—2441no—
437C23—2441no—
438C23#1451973412yes—
439C23#1451975011yescharacteristic
440C23#1451975011yescharacteristic
441C23#1451977212yes—
442C23C2×#1175822211yescharacteristic
443Q8—1222no—
444Q8—1222no—
445Q8—2441no—
446Q8—2441no—
447D4—1222no—
448D4—1222no—
449D4—2444no—
450D4—2444no—
451D4—2441no—
452D4—2441no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—1222no—
456C2×C4—1224no—
457C2×C4—2444no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2444no—
461C2×C4—2444no—
462C2×C4#1451973412yes—
463C2×C4#1451973712yes—
464C2×C4#1451973712yes—
465C2×C4#1451973912yes—
466C2×C4#1451973912yes—
467C2×C4#1451975011yescharacteristic
468C2×C4#1451975011yescharacteristic
469C2×C4#1451976212yes—
470C2×C4#1451976212yes—
471C2×C4#1451976312yes—
472C2×C4#1451976312yes—
473C2×C4#1451977212yes—
474C2×C4C22×C4×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C2×#1175822411yescharacteristic
478C2×C4C2×#1175822211yescharacteristic
479C2×C4C2×#1175822611yescharacteristic
480C8—24no—
481C8—42no—
482C8—1224no—
483C8—1222no—
484C8—1222no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—4882no—
491C22—4882no—
492C22—4882no—
493C22—2442no—
494C22#1683965112yes—
495C22#1683965112yes—
496C22C2×#1451973411yescharacteristic
497C22C2×#1451975011yescharacteristic
498C22C2×#1451977211yescharacteristic
499C4—82no—
500C4—2442no—
501C4—2444no—
502C4#1683965112yes—
503C4#1683965112yes—
504C4C2×#1451973711yescharacteristic
505C4C2×#1451973911yescharacteristic
506C4C2×#1451976211yescharacteristic
507C4C2×#1451976311yescharacteristic
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—4882no—
513C2C2×#1683965111yescharacteristic
514C2#2145553812yes—
subgroupG/H
centerC2×C4#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#21075
LCSG ▷ C2×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. #21454408C2semidirect
5. C2×#16839720C2semidirect
6. #21446479C2semidirect
7. #21446472C2semidirect
8. C2×#16839526C2semidirect
9. C61×#1742C2semidirect
26. #16839744C4semidirect
76. #14519695C2×C4semidirect
77. #14519693C2×C4semidirect
258. #1742D61semidirect
295. C61#21075semidirect

Maximal subgroups

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

subgroupindex
#214544082
#214543392
C2×#168397562
C2×#168397442
C2×#168397202
#214464792
#214464722
C2×#168395262
C61×#17422
#2107561