Finite Group Databasebeta

← #29708558

Subgroups — #29708558

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
1#21454409C211yesmaximal, characteristic, complemented
2C2×#16839788C211yesmaximal, characteristic, complemented
3#21454339C211yesmaximal, characteristic
4C2×#16839759C211yesmaximal, characteristic
5C2×#16839744C211yesmaximal, characteristic
6#21447181C214yesmaximal, complemented
7#21447174C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C22×#108×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839804—22no—
15#16839804C2214yes—
16#16839800—22no—
17#16839788—22no—
18#16839788C2214yes—
19C2×#14519706—22no—
20#16839772C2212yes—
21#16839768C2212yes—
22#16839764—22no—
23#16839759—22no—
24#16839759—22no—
25#16839759—22no—
26C2×#14519702C2212yes—
27C2×#14519699C2212yes—
28#16839744—24no—
29C2×#14519695C2212yes—
30C2×#14519693C2212yes—
31#16839530C2214yes—
32C22×M16×C61C2211yescharacteristic
33C22×M16×C61C2211yescharacteristic
34C23×C8×C61C2211yescharacteristic
35C2×#108×C61—24no—
36C2×#108×C61C2214yes—
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×#11758194—22no—
52C2×#11758194C2312yes—
53C22×#92418—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736D412yes—
61#14519736C2312yes—
62C2×C4⋊Dic61D412yes—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68#14519711—24no—
69#14519709—24no—
70#14519706—28no—
71#14519702—24no—
72#14519702—24no—
73#14519699—24no—
74#14519699—24no—
75#14519695—24no—
76#14519693—24no—
77C23×C4×C61C2311yescharacteristic
78C2×M16×C61—21no—
79C2×M16×C61—22no—
80C2×M16×C61—21no—
81C2×M16×C61—22no—
82C2×M16×C61—22no—
83C2×M16×C61D414yes—
84C2×M16×C61C2312yes—
85C2×M16×C61C2312yes—
86C22×C8×C61—22no—
87C22×C8×C61—21no—
88C22×C8×C61—22no—
89C22×C8×C61—21no—
90C22×C8×C61D412yes—
91C22×C8×C61D412yes—
92C22×C8×C61C2312yes—
93#108×C61—28no—
94C24×C61#4111yescharacteristic
95C23×D61—21no—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×Q812yes—
102C22×C4×C61#4111yescharacteristic
103C22×C4×C61#4112yes—
104C22×C4×C61#4112yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—24no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—42no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—22no—
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—24no—
138M16×C61—22no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61—42no—
142M16×C61C2×D414yes—
143C2×C8×C61—24no—
144C2×C8×C61—24no—
145C2×C8×C61—24no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—42no—
151C2×C8×C61C2×D414yes—
152C22⋊Dic61—22no—
153C22⋊Dic61—22no—
154C22⋊Dic61—41no—
155C22⋊Dic61—41no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C2×D414yes—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164#11758194—22no—
165#11758194—22no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—42no—
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#12112yes—
180C2×C4×C61#12112yes—
181C2×C4×C61#12112yes—
182C2×C4×C61#12112yes—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61C22⋊Q812yes—
186C2×C4×C61C22⋊Q812yes—
187C2×C4×C61C22⋊Q812yes—
188C2×C4×C61C22⋊Q812yes—
189C2×C4×C61C22⋊Q812yes—
190C2×C4×C61#12612yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×Q811yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C22⋊D61—44no—
198C22⋊D61—44no—
199C2×Dic61—24no—
200C2×Dic61—22no—
201C2×Dic61—44no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—82no—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210C4×D61—44no—
211C4×D61—44no—
212Dic122—22no—
213Dic122—22no—
214Dic122—41no—
215Dic122—41no—
216C8×C61—24no—
217C8×C61—24no—
218C8×C61—42no—
219#92418—42no—
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#39612yes—
236C4×C61#39612yes—
237C4×C61C2×#12111yescharacteristic
238C4×C61C2×#12111yescharacteristic
239C4×C61C2×C22⋊Q811yescharacteristic
240C4×C61C2×C22⋊Q811yescharacteristic
241Dic61—42no—
242Dic61—44no—
243Dic61—82no—
244C22×#108D6111yescharacteristic, complemented
245#2911—611no—
246C2×#424—611no—
247#2896—611no—
248C2×#416—611no—
249C2×#413—611no—
250#2064—614no—
251#2063—614no—
252#1742—611no—
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×C8C2×D6111yescharacteristic
265C2×#117—611no—
266C2×#117—611no—
267C2×#108—24no—
268C2×#108C2×D6114yes—
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#343—614no—
287C61#2360111yescharacteristic, complemented, Sylow
288C2×#41—614no—
289C2×#41—1222no—
290C2×#41—1221no—
291C2×#41—1221no—
292C2×#41—1222no—
293C2×#41—1221no—
294C2×#41—1221no—
295C2×#41—1222no—
296C22×Q8—611no—
297C22×D4—611no—
298C22×D4—1221no—
299C23×C4—1221no—
300C23×C4C22×D6111yescharacteristic
301C2×M16—21no—
302C2×M16—22no—
303C2×M16—21no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—612no—
307C2×M16—1221no—
308C2×M16—1222no—
309C2×M16—1222no—
310C2×M16C22⋊D6114yes—
311C2×M16C22×D6112yes—
312C2×M16C22×D6112yes—
313C22×C8—22no—
314C22×C8—21no—
315C22×C8—22no—
316C22×C8—21no—
317C22×C8—1221no—
318C22×C8D24412yes—
319C22×C8D24412yes—
320C22×C8C22×D6112yes—
321#117—612no—
322#117—612no—
323#117—1221no—
324#117—1221no—
325#117—1221no—
326#117—1221no—
327#117—1221no—
328#117—1221no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C42—1221no—
334C2×C42—1221no—
335#108—28no—
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#1175822611yescharacteristic
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—1221no—
372C22×C4—1222no—
373C22×C4—1224no—
374C22×C4—1222no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2442no—
378C22×C4C2×D24411yescharacteristic
379C22×C4C2×D24411yescharacteristic
380C22×C4#1175822212yes—
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822412yes—
384C22×C4Q8×D6112yes—
385C22×C4#1175822611yescharacteristic
386C22×C4C2×C22⋊D6112yes—
387C22×C4C23×D6111yescharacteristic
388M16—22no—
389M16—24no—
390M16—22no—
391M16—22no—
392M16—42no—
393M16—1222no—
394M16—1222no—
395M16—2442no—
396M16C2×C22⋊D6114yes—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—42no—
405C2×C8—1222no—
406C2×C8—2442no—
407C2×C8C2×D24414yes—
408C4⋊C4—614no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—1222no—
417C42—1222no—
418C42—2441no—
419C42—2441no—
420C23—42no—
421C23—1222no—
422C23—4882no—
423C23—2441no—
424C23—2441no—
425C23#1451975211yescharacteristic
426C23#1451975211yescharacteristic
427C23#1451975312yes—
428C23#1451977212yes—
429C23C2×#1175822611yescharacteristic
430Q8—1222no—
431Q8—1222no—
432Q8—2441no—
433Q8—2441no—
434D4—1222no—
435D4—1222no—
436D4—2444no—
437D4—2444no—
438D4—2441no—
439D4—2441no—
440C2×C4—1224no—
441C2×C4—1222no—
442C2×C4—4882no—
443C2×C4—2442no—
444C2×C4—2444no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2444no—
448C2×C4—2444no—
449C2×C4#1451974912yes—
450C2×C4#1451974912yes—
451C2×C4#1451975112yes—
452C2×C4#1451975112yes—
453C2×C4#1451975211yescharacteristic
454C2×C4#1451975211yescharacteristic
455C2×C4#1451975312yes—
456C2×C4#1451975512yes—
457C2×C4#1451975512yes—
458C2×C4#1451975712yes—
459C2×C4#1451975712yes—
460C2×C4#1451977212yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C22×D24411yescharacteristic
464C2×C4C2×Q8×D6111yescharacteristic
465C2×C4C2×#1175822411yescharacteristic
466C2×C4C2×#1175822211yescharacteristic
467C8—24no—
468C8—24no—
469C8—42no—
470C8—2442no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—4882no—
477C22—4882no—
478C22—4882no—
479C22—2442no—
480C22#1683968412yes—
481C22#1683968412yes—
482C22C2×#1451975211yescharacteristic
483C22C2×#1451975311yescharacteristic
484C22C2×#1451977211yescharacteristic
485C4—4882no—
486C4—2442no—
487C4—2444no—
488C4#1683968412yes—
489C4#1683968412yes—
490C4C2×#1451974911yescharacteristic
491C4C2×#1451975111yescharacteristic
492C4C2×#1451975511yescharacteristic
493C4C2×#1451975711yescharacteristic
494C2—21no—
495C2—21no—
496C2—22no—
497C2—22no—
498C2—4882no—
499C2C2×#1683968411yescharacteristic
500C2#2145554012yes—
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#108×C61C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
hypercenterC22×#108D61
hypocenterC61#23601
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
1. #21454409C2semidirect
2. C2×#16839788C2semidirect
6. #21447181C2semidirect
7. #21447174C2semidirect
8. #21445557C2semidirect
9. C22×#108×C61C2semidirect
244. C22×#108D61semidirect
287. C61#23601semidirect

Maximal subgroups

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

subgroupindex
#214544092
C2×#168397882
#214543392
C2×#168397592
C2×#168397442
#214471812
#214471742
#214455572
C22×#108×C612
#2360161