Finite Group Databasebeta

← #10822749

Subgroups — #10822749

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5526C211yesmaximal, characteristic
2C2×#5120C211yesmaximal, characteristic, complemented
3C2×#5129C211yesmaximal, characteristic
4C2×#4998C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic
6C22×#810C211yesmaximal, characteristic, complemented
7C3×#1736C211yesmaximal, characteristic, complemented
8#68825C214yesmaximal, complemented
9#68824C214yesmaximal, complemented
10#23487—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×#926C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18#5526—24no—
19#5526—24no—
20C2×#897C2214yes—
21C2×#886—22no—
22C2×#875—21no—
23C2×#869—21no—
24#5213—22no—
25#5129—24no—
26#5120—24no—
27C2×(C2×D3)⋊C8C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29C2×#816C2214yes—
30#4998—24no—
31#4988—21no—
32#4987C2214yes—
33#4973—21no—
34C2×#810—24no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#808C2212yes—
38C2×#807C2212yes—
39C3×#340C2214yes—
40C2×#536—31no—
41C2×#427—31no—
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1736D311yescharacteristic, complemented
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67D412yes—
52C23×Dic3—21no—
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C22×D12D411yescharacteristic
60C22×C4×D3C2312yes—
61C22×Dic6—22no—
62C22×Dic6—22no—
63C22×Dic6—22no—
64C22×Dic6—22no—
65C22×Dic6—21no—
66C22×Dic6—22no—
67C22×Dic6D411yescharacteristic
68C22×Dic6D414yes—
69C22×Dic6C2312yes—
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117D414yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×#217—21no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86#926—22no—
87#926—22no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3C2312yes—
90C2×C4×Dic3—21no—
91C22×C3⋊C8—22no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8D414yes—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96#897—28no—
97#897—28no—
98#886—216no—
99#875—28no—
100#869—28no—
101C3⋊#95—24no—
102(C2×D3)⋊C8—24no—
103(C2×D3)⋊C8—24no—
104#816—28no—
105#810—24no—
106#810—24no—
107#808—24no—
108#807—24no—
109C22×#41—31no—
110C23×Q8—31no—
111C22×M16C2×D311yescharacteristic
112C23×C8—31no—
113#536—64no—
114#536—64no—
115C2×C4⋊Q8—61no—
116C2×#124—61no—
117C2×#123—62no—
118C2×C22⋊Q8—34no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4C2×D311yescharacteristic
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#419—64no—
126C2×#103—34no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—64no—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340C2×D314yes—
138C24×C3C2×D411yescharacteristic
139C23×D3C2×D412yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—42no—
143C22×C3×C4—82no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4#4112yes—
149C22×C3×C4#4112yes—
150C22×C3×C4#4112yes—
151C22×C3×C4C2411yescharacteristic
152C2×#67—24no—
153C2×#67—24no—
154C2×#67—24no—
155C22×Dic3—24no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—42no—
159C22×Dic3—42no—
160C22×Dic3C2×D412yes—
161#235—44no—
162#235—44no—
163#235—44no—
164#235—44no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×Dic6—28no—
175C2×Dic6—28no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—24no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—42no—
186C2×Dic6—44no—
187C2×Dic6—42no—
188C2×Dic6—44no—
189C2×Dic6C2×D414yes—
190C3×M16—44no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4C2×D414yes—
199C3×C22⋊C4—24no—
200C3×C42—24no—
201#217—42no—
202(C2×D3)⋊C4—216no—
203(C2×D3)⋊C4—84no—
204C4⋊Dic3—216no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4×Dic3—42no—
208C2×C3⋊C8—28no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8C2×D412yes—
214C2×C3⋊C8C2×D412yes—
215C2×#41—62no—
216C2×#41—62no—
217C2×#41—62no—
218C2×#41—62no—
219C2×#41—62no—
220C2×#41—62no—
221C22×Q8—31no—
222C22×Q8—34no—
223C22×Q8—32no—
224C22×Q8—62no—
225C22×Q8—61no—
226C22×Q8—62no—
227C22×Q8—62no—
228C22×Q8—62no—
229C22×Q8—62no—
230C22×D4—32no—
231C22×D4—31no—
232C23×C4—21no—
233C23×C4—32no—
234C23×C4—61no—
235C23×C4C22×D311yescharacteristic
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C22×C8—34no—
240C22×C8—31no—
241C22×C8—31no—
242C22×C8—62no—
243C22×C8—62no—
244C22×C8C22×D312yes—
245C4⋊Q8—68no—
246#124—68no—
247#123—616no—
248C22⋊Q8—68no—
249C22⋊Q8—68no—
250#117—22no—
251#117—62no—
252#117—62no—
253#117D1214yes—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4C22×D312yes—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—64no—
265C2×C42—21no—
266C2×C42—61no—
267#107—64no—
268#106—64no—
269#103—64no—
270#103—64no—
271#103—68no—
272C22⋊C8—64no—
273C22⋊C8—64no—
274#95—64no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12012yes—
278C23×C3#12112yes—
279C23×C3#12312yes—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—24no—
285C22×D3—24no—
286C2×C3×C4—24no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—44no—
291C2×C3×C4—84no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12114yes—
294C2×C3×C4#12112yes—
295C2×C3×C4#12114yes—
296C2×C3×C4#12314yes—
297C2×C3×C4#12312yes—
298C2×C3×C4#12414yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305#67—48no—
306C2×Dic3—22no—
307C2×Dic3—24no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—48no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316D12—44no—
317C4×D3—42no—
318C4×D3—42no—
319C4×D3—42no—
320C4×D3—42no—
321Dic6—24no—
322Dic6—24no—
323Dic6—42no—
324Dic6—42no—
325Dic6—48no—
326Dic6—44no—
327C3×C8—42no—
328C3⋊C8—22no—
329C3⋊C8—22no—
330C24—32no—
331C24D3×D411yescharacteristic
332#41—124no—
333#41—124no—
334#41—124no—
335#41—124no—
336C2×Q8—34no—
337C2×Q8—64no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—68no—
343C2×Q8—68no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—124no—
349C2×Q8—122no—
350C2×Q8—122no—
351C2×Q8—124no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—32no—
361C22×C4—42no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—82no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4C2×D1212yes—
375C22×C4#23512yes—
376C22×C4#23512yes—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D412yes—
379C22×C4D3×D412yes—
380C22×C4#23912yes—
381C22×C4C23×D311yescharacteristic
382M16—44no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8—68no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—616no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4C2×D1214yes—
401C22⋊C4—24no—
402C22⋊C4—616no—
403C22⋊C4—122no—
404C22⋊C4—244no—
405C42—24no—
406C42—122no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3#39414yes—
412C22×C3C2×#12011yescharacteristic
413C22×C3C2×#12111yescharacteristic
414C22×C3C2×#12311yescharacteristic
415C22×C3#57712yes—
416C22×C3#57812yes—
417C2×D3—44no—
418C2×D3—42no—
419C2×D3—44no—
420C2×D3—42no—
421C2×D3—42no—
422C3×C4—44no—
423C3×C4—82no—
424C3×C4#39414yes—
425C3×C4C2×#12111yescharacteristic
426C3×C4C2×#12111yescharacteristic
427C3×C4C2×#12311yescharacteristic
428C3×C4C2×#12411yescharacteristic
429Dic3—44no—
430Dic3—42no—
431Dic3—82no—
432C23—22no—
433C23—22no—
434C23—62no—
435C23—62no—
436C23—62no—
437C23—64no—
438C23—64no—
439C23#88212yes—
440C23#88412yes—
441C23#89412yes—
442C23C2×D3×D411yescharacteristic
443Q8—64no—
444Q8—64no—
445Q8—128no—
446Q8—124no—
447Q8—122no—
448Q8—122no—
449D4—128no—
450D4—124no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—84no—
461C2×C4—128no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4#87514yes—
472C2×C4#88212yes—
473C2×C4#88412yes—
474C2×C4#88414yes—
475C2×C4#89412yes—
476C2×C4#89414yes—
477C2×C4#89514yes—
478C2×C4C22×D1211yescharacteristic
479C2×C4C2×#23511yescharacteristic
480C2×C4C2×#23511yescharacteristic
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×#23911yescharacteristic
484C8—42no—
485C8—62no—
486C8—62no—
487C2×C3—24no—
488C2×C3C2×#39411yescharacteristic
489C2×C3#301312yes—
490C2×C3#305711yescharacteristic
491C2×C3#305911yescharacteristic
492C2×C3#309712yes—
493D3—82no—
494C22—24no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—124no—
499C22—122no—
500C22—122no—
501C22—124no—
502C22—122no—
503C22#467214yes—
504C22C2×#88211yescharacteristic
505C22C2×#88411yescharacteristic
506C22C2×#89411yescharacteristic
507C22#577512yes—
508C22#577912yes—
509C4—44no—
510C4—82no—
511C4—122no—
512C4—124no—
513C4—242no—
514C4#467214yes—
515C4C2×#87511yescharacteristic
516C4C2×#88411yescharacteristic
517C4C2×#89411yescharacteristic
518C4C2×#89511yescharacteristic
519C3#2348711yescharacteristic, complemented, Sylow
520C2—24no—
521C2—242no—
522C2C2×#467211yescharacteristic
523C2#7888712yes—
524C2#7893111yescharacteristic
525C2#7896411yescharacteristic
526C2#7922012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1736D3
hypocenterC3#23487
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#5120C2semidirect
4. C2×#4998C2semidirect
6. C22×#810C2semidirect
7. C3×#1736C2semidirect
8. #68825C2semidirect
9. #68824C2semidirect
48. #1736D3semidirect
519. C3#23487semidirect

Maximal subgroups

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

subgroupindex
C2×#55262
C2×#51202
C2×#51292
C2×#49982
#821862
C22×#8102
C3×#17362
#688252
#688242
#234873