Finite Group Databasebeta

← #10844697

Subgroups — #10844697

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4841C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#74369C211yesmaximal, characteristic
4#73722C211yesmaximal, characteristic, complemented
5C3×#1948C211yesmaximal, characteristic, complemented
6#70589C212yesmaximal, complemented
7#70574C212yesmaximal
8#69579C211yesmaximal, characteristic, complemented
9C4×#809C211yesmaximal, characteristic
10#68863C212yesmaximal
11#68851C212yesmaximal, complemented
12#25293—31nomaximal, Sylow
13C22×#204C2212yes—
14C2×#897—21no—
15C2×C4×D12C2211yescharacteristic
16C2×#869C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22C2×#816C2212yes—
23C2×Dic3⋊C8C2212yes—
24C2×C8×Dic3C2212yes—
25#4994—22no—
26#4988C2212yes—
27#4987—21no—
28#4984C2212yes—
29#4973C2212yes—
30#4973C2212yes—
31C2×#810C2212yes—
32C2×#810C2212yes—
33C2×#809C2211yescharacteristic
34C2×#809C2211yescharacteristic
35#4936—21no—
36#4929—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4841—24no—
39#4830—24no—
40#4659—22no—
41C3×#340C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#461—31no—
45C2×C8×Q8—31no—
46#2568—31no—
47#2538—31no—
48#2066—32no—
49#2055—32no—
50#1968—32no—
51#1956—31no—
52#1948D311yescharacteristic, complemented
53#1933—32no—
54C4×#102—31no—
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×Dic6D411yescharacteristic
58C22×Dic6C2312yes—
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×Dic3⋊C4D412yes—
76C2×C4×Dic3C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79C2×#204—24no—
80C2×#204—24no—
81C2×#204—42no—
82#897—44no—
83C4×D12—22no—
84C4×D12—22no—
85#869—22no—
86#869—22no—
87C4×Dic6—22no—
88C4×Dic6—22no—
89C3×#106—42no—
90C3×C4⋊C8D414yes—
91C3×#95—21no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#816—24no—
98Dic3⋊C8—24no—
99C8×Dic3—24no—
100#810—24no—
101#810—42no—
102#809—22no—
103#809—22no—
104C4×C3⋊C8—22no—
105C22×SD16—32no—
106C2×C4⋊Q8—31no—
107C2×C22⋊Q8—61no—
108C2×C4×Q8—31no—
109C2×C4×D4—31no—
110#461—64no—
111C8×Q8—64no—
112C2×#106C2×D312yes—
113C2×C4⋊C8—32no—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102—31no—
119C2×#102—31no—
120C2×C4×C8—32no—
121C2×C4×C8—31no—
122#400—62no—
123#396—62no—
124#391—32no—
125#391—32no—
126#391—61no—
127#390—32no—
128#389—32no—
129#384C2×D311yescharacteristic
130#382—61no—
131#380—61no—
132C4×C4⋊C4C2×D311yescharacteristic
133#340—31no—
134#340—31no—
135#340C2×D312yes—
136C23×D3—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—21no—
145C22×Dic3C2×D412yes—
146C22×Dic3C2×D412yes—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12#4112yes—
153C2×C4×D3—42no—
154C2×C4×D3—42no—
155C2×C4×D3—42no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—22no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6SD1614yescomplemented
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D414yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—81no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—82no—
182C4⋊Dic3—22no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—42no—
186C4⋊Dic3—42no—
187C4⋊Dic3—44no—
188Dic3⋊C4—24no—
189C4×Dic3—22no—
190C4×Dic3—22no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8#4112yes—
197#204—48no—
198C22×Q8—31no—
199C22×Q8—32no—
200C22×D4—31no—
201C23×C4—61no—
202C2×SD16—64no—
203C2×SD16—64no—
204C2×SD16—122no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8C22×D312yes—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210C22⋊Q8—124no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×D4—62no—
214C4×D4—62no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—62no—
229C2×C42—21no—
230C2×C42—32no—
231C2×C42C22×D311yescharacteristic
232#106—42no—
233C4⋊C8—64no—
234C4⋊C8D1214yes—
235#103—64no—
236#103—64no—
237#103—122no—
238#102—62no—
239#102—62no—
240C4×C8—64no—
241C4×C8—62no—
242#95—21no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—62no—
248C23×C3C22×D411yescharacteristic
249C22×D3—22no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4#12112yes—
268C2×C3×C4#12112yes—
269C2×C3×C4C2×SD1612yes—
270C2×C3×C4#13512yes—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4H(5,𝔽2)11yescharacteristic
276C2×C3×C42−1+411yescharacteristic
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—41no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3C2×SD1614yes—
289D12—22no—
290D12—22no—
291D12—41no—
292D12—41no—
293C4×D3—84no—
294Dic6—28no—
295Dic6—42no—
296Dic6—42no—
297Dic6—42no—
298Dic6—42no—
299C3×C8—24no—
300C3⋊C8—22no—
301C3⋊C8—41no—
302C24—61no—
303C2×Q8—34no—
304C2×Q8—62no—
305C2×Q8—62no—
306C2×Q8—62no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×D4—32no—
313C2×D4—61no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4—61no—
317C2×D4—61no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—128no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—32no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8C2×D1214yes—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—64no—
349C4⋊C4—62no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—81no—
353C4⋊C4—124no—
354C4⋊C4—122no—
355C4⋊C4—122no—
356C22⋊C4—124no—
357C22⋊C4—122no—
358C22⋊C4—242no—
359C42—42no—
360C42—62no—
361C42—62no—
362C42C2×D1212yes—
363C42#23912yes—
364C22×C3C2×#12111yescharacteristic
365C22×C3#54811yescharacteristic
366C22×C3#55011yescharacteristic
367C22×C3C22×SD1611yescharacteristic
368C22×C3C2×#13511yescharacteristic
369C22×C3#57711yescharacteristic
370C22×C3#57811yescharacteristic
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—42no—
374C2×D3—81no—
375C2×D3—81no—
376C2×D3—81no—
377C2×D3—81no—
378C3×C4—22no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4—81no—
382C3×C4#46514yes—
383C3×C4#52911yescharacteristic
384C3×C4#54211yescharacteristic
385C3×C4#54611yescharacteristic
386C3×C4#54811yescharacteristic
387Dic3—24no—
388Dic3—42no—
389Dic3—82no—
390Dic3—81no—
391C23—62no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23C2×D3×D411yescharacteristic
399Q8—68no—
400Q8—122no—
401Q8—122no—
402Q8—122no—
403Q8—122no—
404D4—62no—
405D4—62no—
406D4—121no—
407D4—121no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—34no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—244no—
432C2×C4#89512yes—
433C2×C4#89512yes—
434C2×C4D3×SD1612yes—
435C2×C4#91612yes—
436C2×C4C22×D1211yescharacteristic
437C2×C4#100411yescharacteristic
438C2×C4C2×#23911yescharacteristic
439C2×C4#100711yescharacteristic
440C2×C4D3×#4111yescharacteristic
441C2×C4D3×#4111yescharacteristic
442C8—24no—
443C8—62no—
444C8—121no—
445C2×C3#242611yescharacteristic
446C2×C3C2×#46511yescharacteristic
447C2×C3#306311yescharacteristic
448C2×C3#335911yescharacteristic
449C2×C3#336011yescharacteristic
450C2×C3#339211yescharacteristic
451C2×C3#339311yescharacteristic
452D3—82no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22C2×#89511yescharacteristic
461C22#559911yescharacteristic
462C22#563311yescharacteristic
463C22C2×D3×SD1611yescharacteristic
464C22C2×#91611yescharacteristic
465C22#577511yescharacteristic
466C22#577911yescharacteristic
467C4—22no—
468C4—64no—
469C4—81no—
470C4—81no—
471C4—82no—
472C4—122no—
473C4—241no—
474C4—242no—
475C4#486414yes—
476C4#555411yescharacteristic
477C4#557611yescharacteristic
478C4#569111yescharacteristic
479C4#573311yescharacteristic
480C3#2529311yescharacteristic, complemented, Sylow
481C2—242no—
482C2#7621811yescharacteristic
483C2C2×#486411yescharacteristic
484C2#7921411yescharacteristic
485C2#8037811yescharacteristic
486C2#8042111yescharacteristic
487C2#8103711yescharacteristic
488C2#8108711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1948D3
hypocenterC3#25293
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
4. #73722C2semidirect
5. C3×#1948C2semidirect
6. #70589C2semidirect
8. #69579C2semidirect
11. #68851C2semidirect
52. #1948D3semidirect
164. C2×Dic6SD16semidirect
480. C3#25293semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
C2×#48302
#743692
#737222
C3×#19482
#705892
#705742
#695792
C4×#8092
#688632
#688512
#252933