Finite Group Databasebeta

← #10818094

Subgroups — #10818094

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#472C211yesmaximal, characteristic
2#83930C212yesmaximal, complemented
3#83823C211yesmaximal, characteristic, complemented
4#83785C211yesmaximal, characteristic, complemented
5#72088C212yesmaximal, complemented
6#72038C211yesmaximal, characteristic, complemented
7#72008C211yesmaximal, characteristic, complemented
8#70702C211yesmaximal, characteristic, complemented
9#70701C211yesmaximal, characteristic, complemented
10#70072C212yesmaximal, complemented
11#70068C212yesmaximal, complemented
12#23046—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5827C2212yes—
17#5798C2212yes—
18C3×#472C418yescomplemented
19C2×C3×#108C2211yescharacteristic
20C2×C3×#100C2212yes—
21C2×#837C2211yescharacteristic
22#5233C2211yescharacteristic
23#5221C2212yes—
24#5215C2212yes—
25C2×#832C2212yes—
26#5164—22no—
27Dic3×SD16—22no—
28#5152—22no—
29#5152—22no—
30Dic3×D8—22no—
31Dic3×D8—21no—
32Dic3×D8—21no—
33#5122C2212yes—
34#5121—22no—
35#5121C2212yes—
36#5121C2212yes—
37#5106—21no—
38#5106C2211yescharacteristic
39C2×#818—21no—
40C2×#818C2211yescharacteristic
41#5101C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4558C2211yescharacteristic
44#4555C2212yes—
45#4555C2211yescharacteristic
46#4554C2212yes—
47C2×#472D311yescharacteristic
48#2948—31no—
49#2946—32no—
50#2913—31no—
51#1970—31no—
52#1952—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1772—31no—
57#1764—31no—
58C2×C3×#41C2311yescharacteristic
59C22×C3×D4C2312yes—
60C3×AGL(1,ℤ/8)—24no—
61C3×AGL(1,ℤ/8)C2×C418yescomplemented
62C2×C3×SD16—22no—
63C2×C3×D8—22no—
64C2×C3×D8—21no—
65C2×C3×D8—21no—
66C2×C3×D8—22no—
67C2×C3×D8C2×C414yescomplemented
68C3×#131—22no—
69C3×#131C2×C414yescomplemented
70C2×C3×M16—21no—
71C2×C3×M16C2311yescharacteristic
72C2×C3×M16C2312yes—
73C22×C3×C8—21no—
74C22×C3×C8C2311yescharacteristic
75C2×#217—42no—
76C2×#217—42no—
77Q8×Dic3—22no—
78D4×Dic3—22no—
79D4×Dic3—22no—
80D4×Dic3—22no—
81D4×Dic3—22no—
82D4×Dic3—22no—
83D4×Dic3—42no—
84#926—22no—
85#926C2311yescharacteristic
86#926C2312yes—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3C2312yes—
89C2×C4×Dic3C2311yescharacteristic
90C2×#208C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C3×#108C2×C414yescomplemented
93C3×#100C2×C418yescomplemented
94#837—22no—
95#835D411yescharacteristic
96#835D411yescharacteristic
97#832—22no—
98#832—22no—
99#832—22no—
100#832—22no—
101#832D411yescharacteristic
102#832D411yescharacteristic
103#818—21no—
104#818—21no—
105#818—22no—
106#818—22no—
107#818D411yescharacteristic
108#818D411yescharacteristic
109#817—22no—
110#817—22no—
111#817D412yes—
112#815—21no—
113#815—22no—
114C8×Dic3—21no—
115C8×Dic3—22no—
116C8×Dic3—22no—
117C2×AGL(1,ℤ/8)C2×D312yes—
118C22×D8C2×D311yescharacteristic
119C2×#131C2×D311yescharacteristic
120#520—32no—
121#518—32no—
122#472Dic318yescomplemented
123#442—62no—
124#442—62no—
125#440—62no—
126C4×SD16—62no—
127C4×D8—62no—
128C4×D8—61no—
129C4×D8—61no—
130C2×#108C2×D311yescharacteristic
131#428—32no—
132#428—32no—
133#428—62no—
134#427—31no—
135#427—61no—
136C2×#107—31no—
137C2×#107—61no—
138C2×#104—31no—
139#418—32no—
140#416—31no—
141C2×#102—32no—
142C2×#100C2×D312yes—
143#405—32no—
144#405—31no—
145C2×C4×C8—31no—
146#390—32no—
147#343—32no—
148#341—31no—
149#339—31no—
150#339—32no—
151C24×C3—22no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41C22×C412yes—
155C3×#41#4112yes—
156C2×C3×Q8C2×D411yescharacteristic
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4C22×C414yes—
164C2×C3×D4C2×D411yescharacteristic
165C22×C3×C4—21no—
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C3×SD16—28no—
173C3×D8—28no—
174C3×D8—24no—
175C3×D8—24no—
176C3×M16—21no—
177C3×M16—22no—
178C3×M16—24no—
179C3×M16C22×C412yes—
180C3×M16C22×C414yes—
181C2×C3×C8—22no—
182C2×C3×C8—21no—
183C2×C3×C8—24no—
184C2×C3×C8C22×C412yes—
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8#4112yes—
188#217—22no—
189#217—21no—
190#217—42no—
191#217—42no—
192#217—42no—
193#217—84no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3C2×D411yescharacteristic
198C4⋊Dic3C2×D412yes—
199C4⋊Dic3C2×D412yes—
200C4⋊Dic3C2×D411yescharacteristic
201C4×Dic3—21no—
202C4×Dic3—22no—
203C4×Dic3—21no—
204C4×Dic3—21no—
205C4×Dic3—42no—
206#208—21no—
207#208—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8C2×D411yescharacteristic
212C2×C3⋊C8C2×D411yescharacteristic
213C2×#41C22×D311yescharacteristic
214C22×D4C22×D312yes—
215AGL(1,ℤ/8)—24no—
216AGL(1,ℤ/8)C2×Dic318yescomplemented
217C2×SD16—22no—
218C2×D8—22no—
219C2×D8—21no—
220C2×D8—21no—
221C2×D8—22no—
222C2×D8C2×Dic314yescomplemented
223#131—22no—
224#131C2×Dic314yescomplemented
225C2×M16—21no—
226C2×M16—31no—
227C2×M16C22×D311yescharacteristic
228C2×M16C22×D312yes—
229C22×C8—21no—
230C22×C8—31no—
231C22×C8C22×D311yescharacteristic
232C4×Q8—62no—
233C4×D4—62no—
234C4×D4—62no—
235C4×D4—62no—
236C4×D4—62no—
237C4×D4—62no—
238C4×D4—122no—
239#117—31no—
240#117—32no—
241#117—62no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—31no—
247#108C2×Dic314yescomplemented
248#107—31no—
249#107—31no—
250#107—62no—
251#107—62no—
252#107—61no—
253#107—61no—
254#106—32no—
255#106—62no—
256#106—62no—
257#104—62no—
258#103—31no—
259#103—31no—
260#102—31no—
261#102—31no—
262#102—62no—
263#102—62no—
264#102—62no—
265#102—62no—
266#100C2×Dic318yescomplemented
267#97—62no—
268#97—61no—
269C4×C8—62no—
270C4×C8—61no—
271C4×C8—62no—
272C23×C3—21no—
273C23×C3—24no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—82no—
277C23×C3—82no—
278C23×C3H(5,𝔽2)11yescharacteristic
279C3×Q8—21no—
280C3×Q8C4×D412yes—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—21no—
286C3×D4—42no—
287C3×D4—42no—
288C3×D4C4×D412yes—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4C23×C411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C42−1+411yescharacteristic
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—82no—
307C2×Dic3—162no—
308C2×Dic3—162no—
309C2×Dic3—162no—
310C3×C8—22no—
311C3×C8—24no—
312C3×C8C4×D412yes—
313C3×C8C4×D412yes—
314C3⋊C8—21no—
315C3⋊C8—21no—
316C3⋊C8—41no—
317C24—22no—
318#41—21no—
319#41—21no—
320#41#23712yes—
321#41C22×Dic312yes—
322C2×Q8D3×D411yescharacteristic
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4D3×D411yescharacteristic
330C2×D4C22×Dic314yes—
331C22×C4—21no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—28no—
339D8—28no—
340D8—24no—
341D8—24no—
342M16—21no—
343M16—22no—
344M16—24no—
345M16—61no—
346M16—61no—
347M16C22×Dic312yes—
348M16C22×Dic314yes—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—24no—
352C2×C8—31no—
353C2×C8—31no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8D3×D411yescharacteristic
358C2×C8D3×D411yescharacteristic
359C2×C8#23712yes—
360C2×C8C22×Dic312yes—
361C4⋊C4—31no—
362C4⋊C4—31no—
363C4⋊C4—32no—
364C4⋊C4—32no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—122no—
368C22⋊C4—61no—
369C22⋊C4—62no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—61no—
375C42—62no—
376C42—61no—
377C42—61no—
378C42—122no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—44no—
385C22×C3—84no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3—162no—
389C22×C3#52011yescharacteristic
390C22×C3D4211yescharacteristic
391C22×C3#55311yescharacteristic
392C3×C4—22no—
393C3×C4C2×C4×D411yescharacteristic
394C3×C4C2×C4×D411yescharacteristic
395C3×C4#54711yescharacteristic
396C3×C4#54811yescharacteristic
397Dic3—42no—
398Dic3—42no—
399Dic3—41no—
400Dic3—42no—
401Dic3—41no—
402Dic3—82no—
403C23—21no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—82no—
408C23—82no—
409C23#100911yescharacteristic
410Q8—21no—
411Q8D4×Dic312yes—
412D4—22no—
413D4—22no—
414D4—24no—
415D4—24no—
416D4—21no—
417D4—42no—
418D4—42no—
419D4D4×Dic312yes—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—62no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—242no—
432C2×C4—482no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C2×D3×D411yescharacteristic
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×#23711yescharacteristic
438C2×C4C2×#23711yescharacteristic
439C2×C4#101011yescharacteristic
440C2×C4C23×Dic311yescharacteristic
441C8—22no—
442C8—24no—
443C8—61no—
444C8—61no—
445C8—121no—
446C8D4×Dic312yes—
447C8D4×Dic312yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—42no—
451C2×C3—84no—
452C2×C3#236111yescharacteristic
453C2×C3#329011yescharacteristic
454C2×C3#329211yescharacteristic
455C22—21no—
456C22—21no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—44no—
461C22—84no—
462C22—82no—
463C22—82no—
464C22—162no—
465C22#560911yescharacteristic
466C22#561811yescharacteristic
467C22#582711yescharacteristic
468C4—22no—
469C4—122no—
470C4—121no—
471C4—122no—
472C4—121no—
473C4—122no—
474C4—242no—
475C4#559811yescharacteristic
476C4#561111yescharacteristic
477C4C2×D4×Dic311yescharacteristic
478C4C2×D4×Dic311yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—42no—
483C2—84no—
484C2#8094011yescharacteristic
485C2#8094111yescharacteristic
486C2#8263611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#472C2
socleC22×C3D42
cosocleC22×C3×C4C24
hypercenterC2×#472D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
2. #83930C2semidirect
3. #83823C2semidirect
4. #83785C2semidirect
5. #72088C2semidirect
6. #72038C2semidirect
7. #72008C2semidirect
8. #70702C2semidirect
9. #70701C2semidirect
10. #70072C2semidirect
11. #70068C2semidirect
18. C3×#472C4semidirect
61. C3×AGL(1,ℤ/8)C2×C4semidirect
67. C2×C3×D8C2×C4semidirect
69. C3×#131C2×C4semidirect
92. C3×#108C2×C4semidirect
93. C3×#100C2×C4semidirect
122. #472Dic3semidirect
216. AGL(1,ℤ/8)C2×Dic3semidirect
222. C2×D8C2×Dic3semidirect
224. #131C2×Dic3semidirect
247. #108C2×Dic3semidirect
266. #100C2×Dic3semidirect
479. C3#23046semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4722
#839302
#838232
#837852
#720882
#720382
#720082
#707022
#707012
#700722
#700682
#230463