Finite Group Databasebeta

← #10818095

Subgroups — #10818095

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×#5160C211yesmaximal, characteristic
2#83823C211yesmaximal, characteristic, complemented
3#79146C211yesmaximal, characteristic, complemented
4#78726C212yesmaximal, complemented
5C3×#1970C211yesmaximal, characteristic, complemented
6#70625C211yesmaximal, characteristic, complemented
7#70195C212yesmaximal, complemented
8#69789C211yesmaximal, characteristic, complemented
9#69779C211yesmaximal, characteristic, complemented
10#69663C212yesmaximal, complemented
11#69642C212yesmaximal, complemented
12#23046—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14#5798C2212yes—
15C2×#932C2212yes—
16C2×#901C2211yescharacteristic
17#5504C2212yes—
18C2×C3×#107C2211yescharacteristic
19C2×C3×#102C2212yes—
20C2×C3×C4×C8C2211yescharacteristic
21C3×#390C2212yes—
22C2×#833C2212yes—
23#5215C2212yes—
24#5160C418yescomplemented
25#5152—22no—
26Dic3×D8—21no—
27Dic3×D8—21no—
28#5113C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#5106C2211yescharacteristic
31#5101C2212yes—
32#5101C2211yescharacteristic
33#4964C2212yes—
34#4964C2212yes—
35C2×#805C2211yescharacteristic
36#4881—22no—
37#4880—21no—
38D3×#107—21no—
39#4785—22no—
40#4758—22no—
41#4757—22no—
42#4756—22no—
43#4552C2211yescharacteristic
44#4530C2212yes—
45#4523C2211yescharacteristic
46#4523C2212yes—
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2913—31no—
51#1970D311yescharacteristic, complemented
52#1952—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1772—31no—
57#1764—31no—
58C22×C3×D4C2312yes—
59C2×#235C2311yescharacteristic
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8—22no—
63C2×C3×D8C2×C414yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—42no—
68C2×C3×C42C2311yescharacteristic
69C2×#217—42no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74C2×#214—22no—
75#932—24no—
76#932C2×C418yescomplemented
77C2×#213—22no—
78C4×#67—42no—
79#926C2312yes—
80#926C2311yescharacteristic
81C2×#208C2311yescharacteristic
82C2×#208C2311yescharacteristic
83C2×#208C2312yes—
84#901—22no—
85#901C2×C414yescomplemented
86C2×ΓL(1,𝔽25)—21no—
87C2×D3×C8—21no—
88#893—22no—
89#892—22no—
90D3×C4⋊C4—22no—
91#887—22no—
92C3×#107—21no—
93C3×#107—21no—
94C3×#102—22no—
95C3×#102—22no—
96C3×C4×C8—22no—
97#833C2×C418yescomplemented
98#832—22no—
99#832—22no—
100#821D411yescharacteristic
101#821D411yescharacteristic
102#819C2×C414yescomplemented
103#818D411yescharacteristic
104#818D411yescharacteristic
105#817D412yes—
106#816D411yescharacteristic
107#816D411yescharacteristic
108#815—21no—
109#815—22no—
110C8×Dic3—21no—
111C8×Dic3—22no—
112#808—22no—
113#808—22no—
114#807—22no—
115#807—22no—
116#805—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8C2×D311yescharacteristic
119C2×#131—31no—
120#520—32no—
121#518—32no—
122#472—38no—
123#442—62no—
124#442—62no—
125#440—62no—
126C4×SD16—62no—
127C4×D8—62no—
128C4×D8—61no—
129C4×D8—61no—
130C2×#108—31no—
131#428—32no—
132#428—32no—
133#428—62no—
134#427—31no—
135#427—61no—
136C2×#107—61no—
137C2×#107C2×D311yescharacteristic
138C2×#104—31no—
139#418—32no—
140#416—31no—
141C2×#102C2×D312yes—
142C2×#100—32no—
143#405—32no—
144#405—31no—
145C2×C4×C8C2×D311yescharacteristic
146#390C2×D312yes—
147#343—32no—
148#341—31no—
149#339—31no—
150#339—32no—
151C24×C3—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4C22×C414yes—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2411yescharacteristic
161C2×#67—21no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164#235—21no—
165#235—21no—
166#235C22×C412yes—
167#235#4112yes—
168C2×D12C2×D411yescharacteristic
169C2×C4×D3—21no—
170C2×C4×D3—42no—
171C2×Dic6C2×D411yescharacteristic
172C3×D8—24no—
173C3×D8—24no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8C22×C412yes—
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8#4112yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C22⋊C4—84no—
185C3×C42—21no—
186C3×C42—21no—
187#217—22no—
188#217—21no—
189#217—42no—
190#217—42no—
191#214—28no—
192#213—28no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3C2×D411yescharacteristic
195C4⋊Dic3C2×D412yes—
196C4⋊Dic3C2×D411yescharacteristic
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—42no—
199C4×Dic3—22no—
200C4×Dic3—21no—
201C4×Dic3—42no—
202#208—21no—
203#208—22no—
204#208—21no—
205#208—21no—
206#208C22×C414yes—
207#208C22×C412yes—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—21no—
211ΓL(1,𝔽25)—24no—
212D3×C8—24no—
213C2×#41—31no—
214C22×D4C22×D312yes—
215AGL(1,ℤ/8)—38no—
216AGL(1,ℤ/8)—64no—
217C2×SD16—62no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—22no—
221C2×D8—62no—
222C2×D8C4×D314yescomplemented
223#131—34no—
224#131—62no—
225C2×M16—31no—
226C2×M16—32no—
227C2×M16—31no—
228C2×M16—61no—
229C22×C8—61no—
230C22×C8C22×D311yescharacteristic
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—62no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—122no—
246C2×C42C22×D311yescharacteristic
247#108—34no—
248#107—21no—
249#107—21no—
250#107—31no—
251#107—31no—
252#107—62no—
253#107—62no—
254#106—32no—
255#106—62no—
256#106—62no—
257#104—62no—
258#103—31no—
259#103—31no—
260#102—22no—
261#102—22no—
262#102—31no—
263#102—31no—
264#102—62no—
265#102—62no—
266#100—38no—
267#97—62no—
268#97—61no—
269C4×C8—22no—
270C4×C8—61no—
271C4×C8—62no—
272C23×C3—24no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—82no—
276C23×C3—82no—
277C23×C3H(5,𝔽2)11yescharacteristic
278C22×D3—21no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—42no—
282C3×D4—42no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—82no—
288C2×C3×C4C23×C411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294#67—22no—
295#67—22no—
296C2×Dic3—21no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—162no—
302C2×Dic3—162no—
303D12—21no—
304D12C4×D412yes—
305C4×D3—22no—
306C4×D3—22no—
307C4×D3—162no—
308Dic6—21no—
309Dic6C4×D412yes—
310C3×C8—21no—
311C3×C8—21no—
312C3×C8C4×D412yes—
313C3×C8C4×D412yes—
314C3⋊C8—22no—
315C3⋊C8—24no—
316C3⋊C8—41no—
317C24—22no—
318#41—32no—
319#41—32no—
320#41—61no—
321#41—61no—
322C2×Q8—31no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—31no—
329C2×D4—61no—
330C2×D4C2×C4×D314yes—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—61no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—24no—
340D8—24no—
341D8—68no—
342M16—32no—
343M16—34no—
344M16—61no—
345M16—62no—
346M16—64no—
347M16—61no—
348M16—61no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—62no—
352C2×C8—61no—
353C2×C8—64no—
354C2×C8—61no—
355C2×C8C2×C4×D312yes—
356C2×C8D3×D411yescharacteristic
357C2×C8D3×D411yescharacteristic
358C2×C8D3×D411yescharacteristic
359C2×C8D3×D411yescharacteristic
360C2×C8#23712yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—122no—
368C22⋊C4—61no—
369C22⋊C4—62no—
370C22⋊C4—84no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—122no—
374C42—21no—
375C42—21no—
376C42—61no—
377C42—62no—
378C42—122no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—44no—
382C22×C3—84no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—162no—
386C22×C3#52011yescharacteristic
387C22×C3D4211yescharacteristic
388C22×C3#55311yescharacteristic
389C2×D3—22no—
390C2×D3—41no—
391C2×D3—41no—
392C3×C4—42no—
393C3×C4—82no—
394C3×C4C2×C4×D411yescharacteristic
395C3×C4C2×C4×D411yescharacteristic
396C3×C4#54711yescharacteristic
397C3×C4#54811yescharacteristic
398Dic3—22no—
399Dic3—42no—
400Dic3—41no—
401Dic3—42no—
402Dic3—41no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—61no—
407C23—82no—
408C23—82no—
409C23#100411yescharacteristic
410Q8—32no—
411Q8—61no—
412D4—24no—
413D4—24no—
414D4—32no—
415D4—42no—
416D4—42no—
417D4—62no—
418D4—62no—
419D4—61no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—82no—
432C2×C4—482no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C2×C4C2×#23711yescharacteristic
439C2×C4D3×#4111yescharacteristic
440C2×C4#101011yescharacteristic
441C8—21no—
442C8—21no—
443C8—62no—
444C8—64no—
445C8—121no—
446C8#67⋊C412yes—
447C8#67⋊C412yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—84no—
451C2×C3#236111yescharacteristic
452C2×C3#329011yescharacteristic
453C2×C3#329211yescharacteristic
454D3—42no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—62no—
459C22—84no—
460C22—82no—
461C22—82no—
462C22—121no—
463C22—121no—
464C22—162no—
465C22#554611yescharacteristic
466C22#561411yescharacteristic
467C22#571711yescharacteristic
468C4—42no—
469C4—62no—
470C4—82no—
471C4—121no—
472C4—122no—
473C4—121no—
474C4—122no—
475C4C2×#67⋊C411yescharacteristic
476C4C4×D3×D411yescharacteristic
477C4#561111yescharacteristic
478C4#567711yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—84no—
483C2—122no—
484C2#7643311yescharacteristic
485C2#8179111yescharacteristic
486C2#8183511yescharacteristic
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1970C2
socleC22×C3D42
prime radicalC2×#5160C2
cosocleC22×C3×C4C24
hypercenter#1970D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. #83823C2semidirect
3. #79146C2semidirect
4. #78726C2semidirect
5. C3×#1970C2semidirect
6. #70625C2semidirect
7. #70195C2semidirect
8. #69789C2semidirect
9. #69779C2semidirect
10. #69663C2semidirect
11. #69642C2semidirect
24. #5160C4semidirect
51. #1970D3semidirect
63. C2×C3×D8C2×C4semidirect
76. #932C2×C4semidirect
85. #901C2×C4semidirect
97. #833C2×C4semidirect
102. #819C2×C4semidirect
222. C2×D8C4×D3semidirect
479. C3#23046semidirect

Maximal subgroups

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

subgroupindex
C2×#51602
#838232
#791462
#787262
C3×#19702
#706252
#701952
#697892
#697792
#696632
#696422
#230463