Finite Group Databasebeta

← #14323716

Subgroups — #14323716

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×Q8×Dic15C211yesmaximal, characteristic, complemented
3C2×#91815C211yesmaximal, characteristic
4#11751330C211yesmaximal, characteristic, complemented
5#11751313C211yesmaximal, characteristic
6#11751085C211yesmaximal, characteristic, complemented
7#11751085C212yesmaximal, complemented
8#11751081C212yesmaximal, complemented
9C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
10#11751052C212yesmaximal
11#11751052C212yesmaximal
12#10614475—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C3×C5×C22⋊Q8C418yescomplemented
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C22⋊C4C2212yes—
19C2×#6719C2211yescharacteristic
20C2×#6719C2211yescharacteristic
21C2×#6719C2212yes—
22Q8×Dic15—24no—
23#91815—24no—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×C4⋊Dic15C2211yescharacteristic
26C2×C4⋊Dic15C2212yes—
27C2×C4⋊Dic15C2211yescharacteristic
28C2×C4⋊Dic15C2211yescharacteristic
29C2×C4⋊Dic15C2212yes—
30C2×C4×Dic15C2211yescharacteristic
31C2×C4×Dic15C2212yes—
32C2×C4×Dic15C2212yes—
33C2×C4×Dic15C2211yescharacteristic
34#91115C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#82744—51nomaximal, Hall
40C2×C5×C22⋊Q8D311yescharacteristic
41C2×Q8×Dic5—31no—
42C2×#4072—31no—
43#65519—31no—
44#65501—31no—
45#65274—31no—
46#65274—32no—
47#65270—32no—
48C4⋊C4×Dic5—31no—
49#65240—32no—
50#65240—32no—
51C24×C3×C5C2311yescharacteristic
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8C2×C414yes—
55C22×C3×C4×C5—22no—
56C22×C3×C4×C5—22no—
57C22×C3×C4×C5C2×C414yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2311yescharacteristic
62C22×Dic15C2312yes—
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C22×Dic15C2312yes—
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C3×C5×C4⋊C4C2×C414yes—
69C3×C5×C4⋊C4C2×C418yes—
70C3×C5×C22⋊C4C2×C418yes—
71#6719—22no—
72#6719—22no—
73#6719—42no—
74C4⋊Dic15—22no—
75C4⋊Dic15—22no—
76C4⋊Dic15—22no—
77C4⋊Dic15—22no—
78C4⋊Dic15—24no—
79C4⋊Dic15—42no—
80C4×Dic15—22no—
81C4×Dic15—24no—
82C4×Dic15—42no—
83C4×Dic15D414yes—
84C2×C3×C22⋊Q8D511yescharacteristic
85C2×Q8×Dic3—51no—
86C2×#926—51no—
87#5229—51no—
88#5212—51no—
89#4984—52no—
90#4984—51no—
91#4980—52no—
92Dic3×C4⋊C4—51no—
93#4951—52no—
94#4951—52no—
95C22×C5×Q8C2×D311yescharacteristic
96C23×C4×C5C2×D311yescharacteristic
97C5×C22⋊Q8Dic318yescomplemented
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C4⋊C4C2×D312yes—
100C2×C5×C22⋊C4C2×D312yes—
101C2×C22⋊Dic5—31no—
102C2×C22⋊Dic5—31no—
103C2×C22⋊Dic5—32no—
104Q8×Dic5—64no—
105#4072—64no—
106C2×C4⋊Dic5—31no—
107C2×C4⋊Dic5—31no—
108C2×C4⋊Dic5—32no—
109C2×C4⋊Dic5—31no—
110C2×C4⋊Dic5—31no—
111C2×C4⋊Dic5—32no—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—32no—
115C2×C4×Dic5—32no—
116#3963—32no—
117#3963—32no—
118#3963—32no—
119#3963—32no—
120#3963—32no—
121#2323—151noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C3×C5×Q8—28no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—28no—
138C2×C3×C4×C5C22×C414yes—
139C2×C3×C4×C5C22×C414yes—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5#4112yes—
144C2×Dic15—21no—
145C2×Dic15—21no—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—21no—
154C2×Dic15—22no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—21no—
159C2×Dic15—22no—
160C2×Dic15C2×D412yes—
161C2×Dic15C2×D412yes—
162C22×C3×Q8C2×D511yescharacteristic
163C23×C3×C4C2×D511yescharacteristic
164C3×C22⋊Q8Dic518yescomplemented
165C2×C3×C4⋊C4C2×D511yescharacteristic
166C2×C3×C4⋊C4C2×D512yes—
167C2×C3×C22⋊C4C2×D512yes—
168C2×#217—51no—
169C2×#217—51no—
170C2×#217—52no—
171Q8×Dic3—104no—
172#926—104no—
173C2×C4⋊Dic3—52no—
174C2×C4⋊Dic3—51no—
175C2×C4⋊Dic3—51no—
176C2×C4⋊Dic3—51no—
177C2×C4⋊Dic3—51no—
178C2×C4⋊Dic3—52no—
179C2×C4×Dic3—51no—
180C2×C4×Dic3—52no—
181C2×C4×Dic3—51no—
182C2×C4×Dic3—52no—
183C3⋊#95—52no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C24×C5C22×D311yescharacteristic
189C2×C5×Q8—22no—
190C2×C5×Q8—22no—
191C2×C5×Q8C2×Dic314yes—
192C22×C4×C5—22no—
193C22×C4×C5—22no—
194C22×C4×C5C2×Dic314yes—
195C22×C4×C5C22×D312yes—
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D311yescharacteristic
199C22×Dic5—31no—
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—32no—
203C22×Dic5—32no—
204C22×Dic5—31no—
205C5×C4⋊C4C2×Dic314yes—
206C5×C4⋊C4C2×Dic318yes—
207C5×C22⋊C4C2×Dic318yes—
208C22⋊Dic5—62no—
209C22⋊Dic5—62no—
210C22⋊Dic5—122no—
211C4⋊Dic5—62no—
212C4⋊Dic5—62no—
213C4⋊Dic5—64no—
214C4⋊Dic5—62no—
215C4⋊Dic5—62no—
216C4⋊Dic5—122no—
217C4×Dic5—34no—
218C4×Dic5—64no—
219C4×Dic5—62no—
220C4×Dic5—122no—
221C2×C22⋊Q8D1511yescharacteristic
222C2×C4×Q8—151no—
223C2×#117—151no—
224#391—151no—
225#389—152no—
226#389—151no—
227#388—152no—
228#388—152no—
229#387—152no—
230#380—151no—
231C4×C4⋊C4—151no—
232C22×C3×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C22×C3×C52−1+411yescharacteristic
246C3×C4×C5—22no—
247C3×C4×C5—24no—
248C3×C4×C5—24no—
249C3×C4×C5C4×D414yes—
250Dic15—22no—
251Dic15—22no—
252Dic15—41no—
253Dic15—42no—
254Dic15—41no—
255Dic15—42no—
256C24×C3C22×D511yescharacteristic
257C2×C3×Q8—22no—
258C2×C3×Q8—22no—
259C2×C3×Q8C2×Dic514yes—
260C22×C3×C4—22no—
261C22×C3×C4—22no—
262C22×C3×C4C2×Dic514yes—
263C22×C3×C4C22×D512yes—
264C22×C3×C4C22×D512yes—
265C22×C3×C4C22×D511yescharacteristic
266C22×C3×C4C22×D511yescharacteristic
267C22×Dic3—51no—
268C22×Dic3—51no—
269C22×Dic3—51no—
270C22×Dic3—52no—
271C22×Dic3—52no—
272C22×Dic3—51no—
273C3×C4⋊C4C2×Dic514yes—
274C3×C4⋊C4C2×Dic518yes—
275C3×C22⋊C4C2×Dic518yes—
276#217—102no—
277#217—102no—
278#217—202no—
279C4⋊Dic3—102no—
280C4⋊Dic3—102no—
281C4⋊Dic3—104no—
282C4⋊Dic3—102no—
283C4⋊Dic3—102no—
284C4⋊Dic3—202no—
285C4×Dic3—54no—
286C4×Dic3—104no—
287C4×Dic3—102no—
288C4×Dic3—202no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—21no—
295C23×C5C22×Dic312yes—
296C23×C5C23×D311yescharacteristic
297C5×Q8—28no—
298C2×C4×C5—22no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—28no—
305C2×C4×C5D3×D412yes—
306C2×C4×C5#23712yes—
307C2×C4×C5C22×Dic314yes—
308C2×C4×C5C22×Dic314yes—
309C2×C4×C5C22×Dic312yes—
310C2×C4×C5C22×Dic312yes—
311C2×Dic5—32no—
312C2×Dic5—32no—
313C2×Dic5—61no—
314C2×Dic5—61no—
315C2×Dic5—61no—
316C2×Dic5—61no—
317C2×Dic5—62no—
318C2×Dic5—61no—
319C2×Dic5—61no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5—61no—
323C2×Dic5—62no—
324C2×Dic5—61no—
325C2×Dic5—62no—
326C2×Dic5—61no—
327C2×Dic5—62no—
328C2×Dic5—61no—
329C22×Q8C2×D1511yescharacteristic
330C23×C4C2×D1511yescharacteristic
331C22⋊Q8Dic1518yescomplemented
332C4×Q8—304no—
333#117—304no—
334C2×C4⋊C4—151no—
335C2×C4⋊C4—151no—
336C2×C4⋊C4—151no—
337C2×C4⋊C4—152no—
338C2×C4⋊C4—151no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4C2×D1511yescharacteristic
341C2×C4⋊C4C2×D1512yes—
342C2×C22⋊C4—151no—
343C2×C22⋊C4—151no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4C2×D1512yes—
346C2×C42—151no—
347C2×C42—151no—
348C2×C42—152no—
349C2×C42—152no—
350#95—152no—
351#95—152no—
352#95—152no—
353#95—152no—
354#95—152no—
355C2×C3×C5—42no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51911yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53611yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#54111yescharacteristic
362C2×C3×C5#54211yescharacteristic
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3C22×Dic512yes—
370C23×C3C23×D511yescharacteristic
371C3×Q8—28no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—28no—
379C2×C3×C4D4×D512yes—
380C2×C3×C4#71112yes—
381C2×C3×C4C22×Dic514yes—
382C2×C3×C4C22×Dic514yes—
383C2×C3×C4C22×Dic512yes—
384C2×C3×C4C22×Dic512yes—
385C2×Dic3—52no—
386C2×Dic3—52no—
387C2×Dic3—101no—
388C2×Dic3—101no—
389C2×Dic3—101no—
390C2×Dic3—101no—
391C2×Dic3—102no—
392C2×Dic3—101no—
393C2×Dic3—101no—
394C2×Dic3—102no—
395C2×Dic3—101no—
396C2×Dic3—102no—
397C2×Dic3—101no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3—101no—
401C2×Dic3—102no—
402C2×Dic3—101no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C2×D3×D411yescharacteristic
411C22×C5C2×#23711yescharacteristic
412C22×C5#100711yescharacteristic
413C22×C5#100711yescharacteristic
414C22×C5#100911yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5C23×Dic311yescharacteristic
417C4×C5—22no—
418C4×C5—24no—
419C4×C5—24no—
420C4×C5D4×Dic314yes—
421Dic5—62no—
422Dic5—62no—
423Dic5—122no—
424Dic5—121no—
425Dic5—122no—
426Dic5—121no—
427C24C22×D1511yescharacteristic
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8C2×Dic1514yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—151no—
434C22×C4—151no—
435C22×C4—151no—
436C22×C4—152no—
437C22×C4—151no—
438C22×C4—152no—
439C22×C4C2×Dic1514yes—
440C22×C4C22×D1512yes—
441C22×C4C22×D1512yes—
442C22×C4C22×D1511yescharacteristic
443C22×C4C22×D1511yescharacteristic
444C4⋊C4—302no—
445C4⋊C4—302no—
446C4⋊C4—304no—
447C4⋊C4—302no—
448C4⋊C4—302no—
449C4⋊C4—602no—
450C4⋊C4C2×Dic1514yes—
451C4⋊C4C2×Dic1518yes—
452C22⋊C4—302no—
453C22⋊C4—302no—
454C22⋊C4—602no—
455C22⋊C4C2×Dic1518yes—
456C42—154no—
457C42—304no—
458C42—302no—
459C42—602no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×D4×D511yescharacteristic
469C22×C3C2×#71111yescharacteristic
470C22×C3#414711yescharacteristic
471C22×C3#414711yescharacteristic
472C22×C3#414911yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3C23×Dic511yescharacteristic
475C3×C4—22no—
476C3×C4—24no—
477C3×C4—24no—
478C3×C4D4×Dic514yes—
479Dic3—102no—
480Dic3—102no—
481Dic3—201no—
482Dic3—202no—
483Dic3—201no—
484Dic3—202no—
485C2×C5—42no—
486C2×C5#562511yescharacteristic
487C2×C5#562811yescharacteristic
488C2×C5#562911yescharacteristic
489C2×C5#564311yescharacteristic
490C2×C5C2×D4×Dic311yescharacteristic
491C2×C5#581211yescharacteristic
492C2×C5#582711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C22×Dic1512yes—
500C23C23×D1511yescharacteristic
501Q8—28no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—152no—
510C2×C4—152no—
511C2×C4—301no—
512C2×C4—301no—
513C2×C4—301no—
514C2×C4—301no—
515C2×C4—301no—
516C2×C4—301no—
517C2×C4—302no—
518C2×C4—302no—
519C2×C4—301no—
520C2×C4—301no—
521C2×C4—302no—
522C2×C4—302no—
523C2×C4—302no—
524C2×C4—302no—
525C2×C4—301no—
526C2×C4—301no—
527C2×C4D4×D1512yes—
528C2×C4#681912yes—
529C2×C4C22×Dic1514yes—
530C2×C4C22×Dic1514yes—
531C2×C4C22×Dic1512yes—
532C2×C4C22×Dic1512yes—
533C2×C3—42no—
534C2×C3#6595911yescharacteristic
535C2×C3#6596211yescharacteristic
536C2×C3#6596311yescharacteristic
537C2×C3#6597711yescharacteristic
538C2×C3C2×D4×Dic511yescharacteristic
539C2×C3#6614611yescharacteristic
540C2×C3#6616111yescharacteristic
541C5#8274411yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C2×D4×D1511yescharacteristic
550C22C2×#681911yescharacteristic
551C22#9209811yescharacteristic
552C22#9209811yescharacteristic
553C22#9210011yescharacteristic
554C22#9210111yescharacteristic
555C22C23×Dic1511yescharacteristic
556C4—22no—
557C4—24no—
558C4—24no—
559C4—302no—
560C4—302no—
561C4—601no—
562C4—601no—
563C4—602no—
564C4—602no—
565C4D4×Dic1514yes—
566C3#1061447511yescharacteristic, complemented, Sylow
567C2—42no—
568C2#1175638411yescharacteristic
569C2#1175638711yescharacteristic
570C2#1175638811yescharacteristic
571C2#1175640211yescharacteristic
572C2C2×D4×Dic1511yescharacteristic
573C2#1175657111yescharacteristic
574C2#1175658611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊Q8C2
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5#2323
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic15C2semidirect
4. #11751330C2semidirect
6. #11751085C2semidirect
7. #11751085C2semidirect
8. #11751081C2semidirect
9. C4⋊C4×Dic15C2semidirect
15. C3×C5×C22⋊Q8C4semidirect
97. C5×C22⋊Q8Dic3semidirect
164. C3×C22⋊Q8Dic5semidirect
331. C22⋊Q8Dic15semidirect
460. C3×C5#2323semidirect
541. C5#82744semidirect
566. C3#10614475semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
C2×Q8×Dic152
C2×#918152
#117513302
#117513132
#117510852
#117510812
C4⋊C4×Dic152
#117510522
#106144753
#827445