Finite Group Databasebeta

← #10829292

Subgroups — #10829292

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×#5543C211yesmaximal, characteristic
2#82375C211yesmaximal, characteristic
3C2×#4851C211yesmaximal, characteristic, complemented
4C2×#4716C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic
6#70486C211yesmaximal, characteristic, complemented
7#69964C212yesmaximal, complemented
8#69936C212yesmaximal, complemented
9#69930C212yesmaximal, complemented
10#69928C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24134—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3—21no—
15C22×C4⋊Dic3C2212yes—
16C2×#924C2212yes—
17C2×#924C2212yes—
18C22×Dic12C2211yescharacteristic
19#5543—24no—
20#5543—24no—
21C2×#890—21no—
22C2×#878—22no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30#5222—21no—
31#5212C2212yes—
32C2×#817C2211yescharacteristic
33C2×#817C2212yes—
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36C2×#816C2211yescharacteristic
37C2×#816C2212yes—
38#4983—21no—
39#4944—22no—
40#4936C2211yescharacteristic
41#4851—24no—
42#4716—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#441—31no—
49#2536—31no—
50#2035—32no—
51#2001—32no—
52#1993—31no—
53#1955D311yescharacteristic
54#1872—32no—
55#1853—32no—
56#1805—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C23×Dic3—22no—
60C22×Dic6—21no—
61C22×Dic6C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74D4×Dic3—28no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83#924—24no—
84#924—24no—
85#924—24no—
86#924—42no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×Dic12—22no—
94C2×Dic12—22no—
95C2×Dic12—22no—
96#890—28no—
97#878—216no—
98#869—22no—
99#869D414yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6D414yes—
104C3×C4⋊C8—22no—
105C3×C22⋊C8—24no—
106C3×C22⋊C8—42no—
107C3×#97—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#817—22no—
112#817—24no—
113#816—22no—
114#816—22no—
115#816—24no—
116#816—42no—
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×Q8—31no—
127C2×C4×D4—61no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4—32no—
130#475—64no—
131#441—64no—
132C2×#106—32no—
133C2×#106—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—31no—
138C2×#103—32no—
139C2×C22⋊C8C2×D312yes—
140C2×C22⋊C8C2×D312yes—
141C2×#97C2×D311yescharacteristic
142#398—61no—
143#394—61no—
144#388—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—28no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—82no—
171C22×Dic3—82no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—24no—
183C2×Dic6—82no—
184C2×Dic6C2×D412yes—
185C2×Dic6#4112yes—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8#4112yes—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—24no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—28no—
200#217—44no—
201#217—44no—
202C4⋊Dic3—28no—
203C4⋊Dic3—24no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3C2×D414yes—
212Dic3⋊C4—28no—
213Dic3⋊C4—24no—
214Dic3⋊C4—44no—
215Dic3⋊C4—42no—
216C4×Dic3—24no—
217C4×Dic3—42no—
218Dic12—28no—
219C22×Q8—31no—
220C22×Q8—31no—
221C22×Q8—61no—
222C22×D4D1211yescharacteristic
223C23×C4—62no—
224C23×C4C22×D312yes—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D312yes—
230C22×C8C22×D311yescharacteristic
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—34no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×Q8—62no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—68no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4D1211yescharacteristic
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4D1212yes—
263C2×C42—61no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—62no—
267#106—64no—
268C4⋊C8—22no—
269#103—62no—
270#103—64no—
271#103—62no—
272#103—122no—
273C22⋊C8—24no—
274C22⋊C8—42no—
275#97—22no—
276#95—62no—
277#95—61no—
278#95—61no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3#13714yes—
286C23×C3C22×D411yescharacteristic
287C3×D4—28no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—24no—
308C2×Dic3—22no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—41no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—88no—
322Dic6—22no—
323Dic6—22no—
324Dic6—22no—
325Dic6—22no—
326Dic6—41no—
327Dic6—41no—
328Dic6—41no—
329Dic6—41no—
330Dic6—84no—
331C3×C8—22no—
332C3×C8—41no—
333C3×C8—42no—
334C24C2×D1212yes—
335C2×Q8—32no—
336C2×Q8—32no—
337C2×Q8—64no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—242no—
349C2×D4—24no—
350C2×D4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—68no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—242no—
365C22×C4—242no—
366C22×C4C2×D1212yes—
367C22×C4C2×D1211yescharacteristic
368C22×C4C2×D1211yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—68no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8#23712yes—
380C4⋊C4—22no—
381C4⋊C4—32no—
382C4⋊C4—34no—
383C4⋊C4—68no—
384C4⋊C4—68no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—61no—
388C4⋊C4—62no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—124no—
394C4⋊C4—122no—
395C22⋊C4—24no—
396C22⋊C4—68no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—64no—
400C42—122no—
401C42D3×D412yes—
402C42D3×D412yes—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C2×#12011yescharacteristic
411C22×C3#54611yescharacteristic
412C22×C3#54711yescharacteristic
413C22×C3C2×#13712yes—
414C22×C3#57712yes—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4#52511yescharacteristic
419C3×C4#53711yescharacteristic
420C3×C4#54811yescharacteristic
421C3×C4#55011yescharacteristic
422Dic3—42no—
423Dic3—42no—
424Dic3—44no—
425Dic3—42no—
426Dic3—81no—
427Dic3—82no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23#90914yes—
435C23C22×D1211yescharacteristic
436Q8—62no—
437Q8—62no—
438Q8—62no—
439Q8—62no—
440Q8—121no—
441Q8—121no—
442Q8—121no—
443Q8—121no—
444Q8—244no—
445D4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—62no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4#88212yes—
473C2×C4#88212yes—
474C2×C4C2×D3×D411yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×#23711yescharacteristic
477C2×C4#100411yescharacteristic
478C2×C4D3×#4111yescharacteristic
479C2×C4#101011yescharacteristic
480C8—22no—
481C8—42no—
482C8—41no—
483C2×C3—44no—
484C2×C3#240711yescharacteristic
485C2×C3#300411yescharacteristic
486C2×C3#300911yescharacteristic
487C2×C3#331612yes—
488C2×C3#334412yes—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C2×#88211yescharacteristic
497C22#555411yescharacteristic
498C22#555511yescharacteristic
499C22C2×#90912yes—
500C22#575112yes—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—122no—
505C4—124no—
506C4—122no—
507C4—122no—
508C4—241no—
509C4—242no—
510C4#560611yescharacteristic
511C4#560811yescharacteristic
512C4#567711yescharacteristic
513C4#571811yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—44no—
516C2#7643511yescharacteristic
517C2#7864511yescharacteristic
518C2#7867111yescharacteristic
519C2#7947012yes—
520C2#7951312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#1955D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
3. C2×#4851C2semidirect
4. C2×#4716C2semidirect
6. #70486C2semidirect
7. #69964C2semidirect
8. #69936C2semidirect
9. #69930C2semidirect
10. #69928C2semidirect
11. #69828C2semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
#823752
C2×#48512
C2×#47162
C3×#19552
#704862
#699642
#699362
#699302
#699282
#698282
#241343