Finite Group Databasebeta

← #10829295

Subgroups — #10829295

Order 768.

1120 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×#4851C211yesmaximal, characteristic, complemented
4#74240C211yesmaximal, characteristic
5C3×#1805C211yesmaximal, characteristic, complemented
6#70471C211yesmaximal, characteristic
7#70273C212yesmaximal, complemented
8#70243C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68878C212yesmaximal, complemented
11#68875C212yesmaximal, complemented
12#24134—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2212yes—
15C22×#216C2211yescharacteristic
16#5579—24no—
17#5579—24no—
18C2×#897C2212yes—
19C2×#896—22no—
20C2×D3×C4⋊C4C2212yes—
21C2×C4×D12—21no—
22C2×C4×D12C2211yescharacteristic
23C2×#870—21no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C3×#382C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2212yes—
33C2×#816C2212yes—
34#5030—24no—
35#4987C2212yes—
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4851—24no—
40#4672—21no—
41#4670—22no—
42#4661—21no—
43C3×#340C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#441—31no—
49#2536—31no—
50#2035—32no—
51#2001—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1853—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12D411yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3C2312yes—
61C22×Dic6—21no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×C4×Q8—22no—
65C3×C4×Q8D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71#946—24no—
72#946—24no—
73C2×#216—22no—
74C2×#216—22no—
75C2×#216—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4D412yes—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—21no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#897—24no—
92#897—42no—
93#896—216no—
94D3×C4⋊C4—24no—
95D3×C4⋊C4—24no—
96C4×D12—28no—
97C4×D12—22no—
98C4×D12—22no—
99#870—28no—
100#869—22no—
101#869D414yes—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C3×C4⋊C8—22no—
105C3×#103—42no—
106C3×#95—21no—
107C3×#95—21no—
108#835—22no—
109C3⋊#95—22no—
110(C2×D3)⋊C8—24no—
111(C2×D3)⋊C8—42no—
112#817—24no—
113#816—24no—
114#810—22no—
115#808—22no—
116#803—22no—
117C22×Q16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C22×C4⋊C4—32no—
130#475—64no—
131#441—64no—
132C2×#106—32no—
133C2×#106—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#103C2×D312yes—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#398—61no—
143#394—61no—
144#388—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340—32no—
149#340C2×D312yes—
150C23×D3C2×D412yes—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8C2×D412yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C2×D12—24no—
168C2×D12—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—28no—
176C2×C4×D3—82no—
177C2×C4×D3—82no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—24no—
184C2×Dic6—82no—
185C2×Dic6#4112yes—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4C2×D412yes—
194C3×C42—24no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#216—28no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—44no—
201(C2×D3)⋊C4—44no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—44no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—42no—
212C4×Dic3—42no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8—31no—
220C22×Q8—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C23×C4—32no—
224C23×C4—62no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8—31no—
229C22×C8—31no—
230C22×C8C22×D312yes—
231C4⋊Q8—34no—
232C4⋊Q8—62no—
233#125—68no—
234C22⋊Q8—616no—
235C22⋊Q8—64no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C4×Q8—22no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×Q8D1214yes—
243C4×D4—68no—
244C4×D4—62no—
245C4×D4—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C22×D312yes—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C42—21no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—64no—
267#106—62no—
268C4⋊C8—22no—
269#103—42no—
270#103—64no—
271#103—62no—
272#103—62no—
273C22⋊C8—64no—
274C22⋊C8—122no—
275#97—62no—
276#95—21no—
277#95—21no—
278#95—62no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3#13714yes—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—41no—
290C3×Q8—41no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4H(5,𝔽2)11yescharacteristic
308C2×C3×C42−1+411yescharacteristic
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320D12—28no—
321C4×D3—42no—
322C4×D3—42no—
323C4×D3—42no—
324C4×D3—42no—
325C4×D3—88no—
326Dic6—22no—
327Dic6—22no—
328Dic6—41no—
329Dic6—41no—
330Dic6—84no—
331C3×C8—42no—
332C3⋊C8—22no—
333C3⋊C8—41no—
334C24—32no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—32no—
341C2×Q8—64no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—242no—
348C2×Q8C2×D1212yes—
349C2×D4—64no—
350C2×D4—62no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—32no—
355C22×C4—68no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—242no—
366C22×C4—242no—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—68no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—32no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—34no—
385C4⋊C4—68no—
386C4⋊C4—68no—
387C4⋊C4—62no—
388C4⋊C4—64no—
389C4⋊C4—62no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—124no—
393C4⋊C4—122no—
394C4⋊C4C2×D1212yes—
395C22⋊C4—68no—
396C22⋊C4—64no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—24no—
400C42—122no—
401C42C2×D1212yes—
402C42D3×D412yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54611yescharacteristic
405C22×C3#54711yescharacteristic
406C22×C3C2×#13712yes—
407C22×C3#57712yes—
408C2×D3—24no—
409C2×D3—24no—
410C2×D3—24no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—42no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—82no—
420C3×C4#52511yescharacteristic
421C3×C4#53711yescharacteristic
422C3×C4#54811yescharacteristic
423C3×C4#55011yescharacteristic
424Dic3—42no—
425Dic3—42no—
426Dic3—44no—
427Dic3—81no—
428C23—34no—
429C23—62no—
430C23—62no—
431C23—62no—
432C23—62no—
433C23—62no—
434C23—62no—
435C23C2×D3×D411yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—62no—
441Q8—62no—
442Q8—121no—
443Q8—121no—
444Q8—244no—
445D4—68no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4#88212yes—
473C2×C4#88212yes—
474C2×C4C22×D1211yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4#100411yescharacteristic
477C2×C4#100711yescharacteristic
478C2×C4D3×#4111yescharacteristic
479C2×C4D3×#4111yescharacteristic
480C8—42no—
481C8—62no—
482C8—121no—
483C2×C3#240711yescharacteristic
484C2×C3#300411yescharacteristic
485C2×C3#300911yescharacteristic
486C2×C3#331612yes—
487C2×C3#334412yes—
488D3—44no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22C2×#88211yescharacteristic
497C22#561211yescharacteristic
498C22#563111yescharacteristic
499C22#577012yes—
500C22D3×#13712yes—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—122no—
507C4—122no—
508C4—124no—
509C4—241no—
510C4#551011yescharacteristic
511C4#552411yescharacteristic
512C4#559911yescharacteristic
513C4#563311yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—124no—
516C2#7401511yescharacteristic
517C2#7883511yescharacteristic
518C2#7895811yescharacteristic
519C2#8062012yes—
520C2#8125512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1805D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
3. C2×#4851C2semidirect
5. C3×#1805C2semidirect
7. #70273C2semidirect
8. #70243C2semidirect
9. #69555C2semidirect
10. #68878C2semidirect
11. #68875C2semidirect
56. #1805D3semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×#50302
C2×#48512
#742402
C3×#18052
#704712
#702732
#702432
#695552
#688782
#688752
#241343