Finite Group Databasebeta

← #10853316

Subgroups — #10853316

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74324C211yesmaximal, characteristic
2#73715C211yesmaximal, characteristic, complemented
3C3×#1948C211yesmaximal, characteristic, complemented
4#70585C212yesmaximal
5#70558C212yesmaximal, complemented
6#70479C211yesmaximal, characteristic
7#70468C211yesmaximal, characteristic, complemented
8#69574C211yesmaximal, characteristic, complemented
9C4×#809C211yesmaximal, characteristic
10#68898C212yesmaximal
11#68799C212yesmaximal, complemented
12#26024—31nomaximal, Sylow
13C2×#892—21no—
14C2×#888C2212yes—
15C2×C4×D12C2211yescharacteristic
16C2×#869—21no—
17C2×C3×#106C2212yes—
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×#821C2212yes—
22C2×Dic3⋊C8C2212yes—
23#4990—22no—
24#4988C2212yes—
25#4987—21no—
26#4984C2212yes—
27Dic3×C4⋊C4C2212yes—
28C2×#809C2211yescharacteristic
29C2×#809C2211yescharacteristic
30#4935C2211yescharacteristic
31C4×C4⋊Dic3C2211yescharacteristic
32#4928—21no—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4656—22no—
35C3×#340C2212yes—
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4529C2211yescharacteristic
40#4529C2211yescharacteristic
41#4529C2212yes—
42#2551—31no—
43#2354—31no—
44#2065—32no—
45#2023—32no—
46#1987—32no—
47#1948D311yescharacteristic, complemented
48#1939—31no—
49#1920—32no—
50#1834—31no—
51#1802—31no—
52C4×#102—31no—
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×Dic6—22no—
56C22×C3×C8C2312yes—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79#892—44no—
80#888—24no—
81#888—42no—
82C4×D12—22no—
83C4×D12Q814yescomplemented
84#869—44no—
85C3×#106—24no—
86C3×C4⋊C8—22no—
87C3×#95—21no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95D412yes—
91#821—42no—
92Dic3⋊C8—42no—
93#809—22no—
94#809—22no—
95C4×C3⋊C8—22no—
96C2×C4⋊Q8—32no—
97C2×C4⋊Q8—61no—
98C2×C4×D4—31no—
99C2×#117—61no—
100C2×#106C2×D312yes—
101C2×C4⋊C8—32no—
102C2×C4⋊C8C2×D311yescharacteristic
103C2×#102—31no—
104C2×#102—31no—
105C2×#102—32no—
106C2×C4×C8—31no—
107#395—62no—
108#390—32no—
109#389—32no—
110#389—61no—
111#388—62no—
112#384—31no—
113#384C2×D311yescharacteristic
114#380—61no—
115C4×C4⋊C4—31no—
116C4×C4⋊C4—32no—
117C4×C4⋊C4C2×D311yescharacteristic
118#340—32no—
119#340—32no—
120#340—32no—
121#340—31no—
122#340—31no—
123#340C2×D312yes—
124#336—32no—
125C23×D3—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3C2×D412yes—
135C22×Dic3C2×D412yes—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12C2×Q812yes—
142C2×C4×D3—24no—
143C2×C4×D3—42no—
144C2×C4×D3—42no—
145C2×Dic6—44no—
146C2×Dic6—44no—
147C2×Dic6—82no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4—81no—
157C3×C4⋊C4#4112yes—
158C3×C4⋊C4#4112yes—
159C3×C42—42no—
160C3×C42C2×Q812yes—
161C3×C42#4112yes—
162(C2×D3)⋊C4—24no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—22no—
166C4⋊Dic3—21no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—44no—
170C4⋊Dic3—42no—
171C4⋊Dic3—81no—
172C4⋊Dic3SD1614yes—
173C4⋊Dic3C2×Q812yes—
174Dic3⋊C4—44no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3—44no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C22×Q8—62no—
185C22×D4—31no—
186C23×C4—61no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8C22×D312yes—
190C4⋊Q8—64no—
191C4⋊Q8—124no—
192C4⋊Q8—122no—
193C4×D4—34no—
194C4×D4—62no—
195#117—124no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4C22×D312yes—
207C2×C4⋊C4C22×D311yescharacteristic
208C2×C4⋊C4C22×D311yescharacteristic
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C42—21no—
213C2×C42—32no—
214C2×C42—62no—
215C2×C42C22×D311yescharacteristic
216#106—24no—
217C4⋊C8—22no—
218C4⋊C8—122no—
219#102—62no—
220#102—62no—
221#102—122no—
222C4×C8—62no—
223#95—21no—
224#95—32no—
225#95—62no—
226#95—62no—
227C23×C3C22×D411yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4C22⋊Q812yes—
247C2×C3×C4C22⋊Q812yes—
248C2×C3×C4C2×SD1612yes—
249C2×C3×C4AGL(1,ℤ/8)12yes—
250C2×C3×C4C22×Q811yescharacteristic
251C2×C3×C4C2×#4111yescharacteristic
252C2×C3×C4C2×#4111yescharacteristic
253C2×C3×C4C2×#4111yescharacteristic
254C2×C3×C4H(5,𝔽2)11yescharacteristic
255C2×C3×C42−1+411yescharacteristic
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3C2×SD1614yes—
268D12—22no—
269D12—22no—
270D12—41no—
271D12—41no—
272C4×D3—48no—
273Dic6—88no—
274C3×C8—42no—
275C3⋊C8—41no—
276C3⋊C8—41no—
277C24—61no—
278C2×Q8—124no—
279C2×Q8—124no—
280C2×Q8—242no—
281C2×D4—32no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—61no—
293C22×C4—64no—
294C22×C4—62no—
295C22×C4—122no—
296C22×C4—122no—
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—61no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—34no—
314C4⋊C4—32no—
315C4⋊C4—42no—
316C4⋊C4—62no—
317C4⋊C4—61no—
318C4⋊C4—62no—
319C4⋊C4—62no—
320C4⋊C4—81no—
321C4⋊C4—124no—
322C4⋊C4—124no—
323C4⋊C4—122no—
324C4⋊C4—241no—
325C4⋊C4#23712yes—
326C4⋊C4#23912yes—
327C22⋊C4—64no—
328C22⋊C4—124no—
329C22⋊C4—242no—
330C42—42no—
331C42—62no—
332C42—62no—
333C42—124no—
334C42#23512yes—
335C42D3×Q812yes—
336C22×C3C2×C22⋊Q811yescharacteristic
337C22×C3#54611yescharacteristic
338C22×C3#54811yescharacteristic
339C22×C3C22×SD1611yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3#57611yescharacteristic
342C22×C3#57811yescharacteristic
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—42no—
346C2×D3—81no—
347C2×D3—81no—
348C2×D3—81no—
349C2×D3—81no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—82no—
354C3×C4#47614yes—
355C3×C4#53311yescharacteristic
356C3×C4#54411yescharacteristic
357C3×C4#55511yescharacteristic
358C3×C4#55611yescharacteristic
359Dic3—24no—
360Dic3—42no—
361Dic3—42no—
362Dic3—82no—
363C23—62no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—121no—
369C23—121no—
370C23C2×D3×D411yescharacteristic
371Q8—248no—
372D4—62no—
373D4—62no—
374D4—121no—
375D4—121no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—34no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—128no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4#89612yes—
401C2×C4#89612yes—
402C2×C4D3×SD1612yes—
403C2×C4#91412yes—
404C2×C4C2×#23511yescharacteristic
405C2×C4C2×#23711yescharacteristic
406C2×C4C2×D3×Q811yescharacteristic
407C2×C4C2×#23911yescharacteristic
408C2×C4#100911yescharacteristic
409C2×C4#101011yescharacteristic
410C8—42no—
411C8—121no—
412C8—121no—
413C2×C3#243911yescharacteristic
414C2×C3C2×#47611yescharacteristic
415C2×C3#308311yescharacteristic
416C2×C3#329811yescharacteristic
417C2×C3#332511yescharacteristic
418C2×C3#336211yescharacteristic
419C2×C3#336711yescharacteristic
420D3—82no—
421C22—122no—
422C22—122no—
423C22—122no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22C2×#89611yescharacteristic
429C22#559811yescharacteristic
430C22#563011yescharacteristic
431C22C2×D3×SD1611yescharacteristic
432C22C2×#91411yescharacteristic
433C22#577411yescharacteristic
434C22#577911yescharacteristic
435C4—22no—
436C4—42no—
437C4—42no—
438C4—64no—
439C4—82no—
440C4—122no—
441C4—122no—
442C4—242no—
443C4#487014yes—
444C4#556311yescharacteristic
445C4#558411yescharacteristic
446C4#568911yescharacteristic
447C4#572811yescharacteristic
448C3#2602411yescharacteristic, complemented, Sylow
449C2—242no—
450C2#7622311yescharacteristic
451C2C2×#487011yescharacteristic
452C2#7924311yescharacteristic
453C2#8036911yescharacteristic
454C2#8041611yescharacteristic
455C2#8102211yescharacteristic
456C2#8107911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1948D3
hypocenterC3#26024
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. #73715C2semidirect
3. C3×#1948C2semidirect
5. #70558C2semidirect
7. #70468C2semidirect
8. #69574C2semidirect
11. #68799C2semidirect
47. #1948D3semidirect
83. C4×D12Q8semidirect
448. C3#26024semidirect

Maximal subgroups

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

subgroupindex
#743242
#737152
C3×#19482
#705852
#705582
#704792
#704682
#695742
C4×#8092
#688982
#687992
#260243