Finite Group Databasebeta

← #10818089

Subgroups — #10818089

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×#4897C211yesmaximal, characteristic
2#79201C212yesmaximal, complemented
3#79146C211yesmaximal, characteristic, complemented
4#78561C211yesmaximal, characteristic, complemented
5C3×#1952C211yesmaximal, characteristic, complemented
6#70702C211yesmaximal, characteristic, complemented
7#70674C212yesmaximal, complemented
8#70628C211yesmaximal, characteristic, complemented
9#69887C211yesmaximal, characteristic, complemented
10#69005C212yesmaximal, complemented
11#68983C212yesmaximal, complemented
12#23046—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5521C2212yes—
17#5504C2212yes—
18C2×C3×#108C2211yescharacteristic
19C3×#428C2212yes—
20C2×C3×#107C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C2×#825C2211yescharacteristic
23#5131C2212yes—
24C2×#823C2212yes—
25#5122C2212yes—
26#5106C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5002C2211yescharacteristic
29#4988C2212yes—
30C2×#809C2212yes—
31#4964C2212yes—
32#4897C418yescomplemented
33#4889—22no—
34#4881—22no—
35#4880—21no—
36D3×#107—21no—
37#4872—22no—
38#4869—22no—
39#4852—22no—
40#4711—22no—
41C4×D24—21no—
42C4×D24—21no—
43C3×#343C2212yes—
44#4558C2211yescharacteristic
45#4555C2212yes—
46#4532C2211yescharacteristic
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2913—31no—
51#1970—31no—
52#1952D311yescharacteristic, complemented
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1772—31no—
57#1764—31no—
58C2×#235C2311yescharacteristic
59C22×D12C2312yes—
60C2×C3×M16C2311yescharacteristic
61C2×C3×M16C2312yes—
62C22×C3×C8C2311yescharacteristic
63C3×#117C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C4×#67—42no—
66C2×(C2×D3)⋊C4—42no—
67#926C2312yes—
68C2×C4×Dic3C2311yescharacteristic
69C2×#208C2311yescharacteristic
70C22×C3⋊C8C2311yescharacteristic
71#908—24no—
72#908C2×C418yescomplemented
73C2×D24—22no—
74C2×D24—21no—
75C2×D24—21no—
76C2×D24—22no—
77C2×D24C2×C414yescomplemented
78C2×#204—22no—
79#901—22no—
80#901C2×C414yescomplemented
81C2×ΓL(1,𝔽25)—21no—
82C2×D3×C8—21no—
83#893—22no—
84#893—22no—
85#893—22no—
86#892—22no—
87D3×C4⋊C4—22no—
88#887—22no—
89D3×C22⋊C4—42no—
90C4×D12—22no—
91C4×D12—22no—
92C3×#108C2×C414yescomplemented
93C3×#107—21no—
94C3×#107—21no—
95C3×#107—22no—
96C3×#106—22no—
97C3×#97—21no—
98C3×C4×C8—21no—
99#825—22no—
100#823C2×C418yescomplemented
101#821—22no—
102#821—22no—
103#818D411yescharacteristic
104#818D411yescharacteristic
105#817D412yes—
106#815—22no—
107C8×Dic3—22no—
108C8×Dic3—22no—
109#810D411yescharacteristic
110#810D411yescharacteristic
111#809—22no—
112#809—22no—
113#809D411yescharacteristic
114#809D411yescharacteristic
115#808—22no—
116#807—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
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—61no—
128C4×D8—61no—
129C4×D8—62no—
130C2×#108C2×D311yescharacteristic
131#428—32no—
132#428—62no—
133#428C2×D312yes—
134#427—31no—
135#427—61no—
136C2×#107—61no—
137C2×#107C2×D311yescharacteristic
138C2×#104—31no—
139#418—32no—
140#416—31no—
141C2×#102—32no—
142C2×#100—32no—
143#405—32no—
144#405C2×D311yescharacteristic
145C2×C4×C8—31no—
146#390—32no—
147#343C2×D312yes—
148#341—31no—
149#339—32no—
150#339—31no—
151C23×D3—22no—
152C22×C3×C4—22no—
153C22×C3×C4C2411yescharacteristic
154C2×#67—21no—
155C22×Dic3—21no—
156#235—21no—
157#235—21no—
158#235C22×C412yes—
159#235#4112yes—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12C22×C414yes—
166C2×D12C2×D411yescharacteristic
167C2×C4×D3—21no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×Dic6C2×D411yescharacteristic
172C3×M16—22no—
173C3×M16—21no—
174C3×M16—21no—
175C3×M16C22×C414yes—
176C2×C3×C8—22no—
177C2×C3×C8—21no—
178C2×C3×C8C22×C412yes—
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8#4112yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4C2×D411yescharacteristic
185C3×C4⋊C4C2×D411yescharacteristic
186C3×C22⋊C4—21no—
187C3×C42—21no—
188#217—22no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—84no—
193C4⋊Dic3C2×D412yes—
194C4⋊Dic3C2×D412yes—
195Dic3⋊C4—42no—
196C4×Dic3—22no—
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3—42no—
200#208—21no—
201#208C22×C412yes—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8C2×D411yescharacteristic
206C2×C3⋊C8C2×D411yescharacteristic
207D24—28no—
208D24—24no—
209D24—24no—
210#204—28no—
211ΓL(1,𝔽25)—24no—
212D3×C8—24no—
213C2×#41—31no—
214C22×D4—32no—
215AGL(1,ℤ/8)—38no—
216AGL(1,ℤ/8)—64no—
217C2×SD16—62no—
218C2×D8—34no—
219C2×D8—62no—
220C2×D8—61no—
221C2×D8—61no—
222C2×D8—62no—
223#131—34no—
224#131—62no—
225C2×M16—31no—
226C2×M16—61no—
227C2×M16C22×D311yescharacteristic
228C2×M16C22×D312yes—
229C22×C8—31no—
230C22×C8—61no—
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—32no—
240#117—62no—
241#117C22×D311yescharacteristic
242C2×C4⋊C4—62no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—31no—
247#108C4×D314yescomplemented
248#107—21no—
249#107—21no—
250#107—22no—
251#107—31no—
252#107—31no—
253#107—62no—
254#106—22no—
255#106—32no—
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#100—38no—
267#97—21no—
268#97—62no—
269C4×C8—21no—
270C4×C8—62no—
271C4×C8—62no—
272C23×C3H(5,𝔽2)11yescharacteristic
273C22×D3—21no—
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—82no—
278C22×D3—82no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—42no—
282C2×C3×C4—82no—
283C2×C3×C4C23×C411yescharacteristic
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C42−1+411yescharacteristic
289#67—22no—
290#67—22no—
291C2×Dic3—21no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297D12—24no—
298D12—24no—
299D12—21no—
300D12—42no—
301D12—42no—
302D12C4×D412yes—
303C4×D3—22no—
304C4×D3—22no—
305C4×D3—162no—
306C4×D3—162no—
307C4×D3—162no—
308Dic6—21no—
309Dic6C4×D412yes—
310C3×C8—24no—
311C3×C8—41no—
312C3×C8C4×D412yes—
313C3×C8C4×D412yes—
314C3⋊C8—22no—
315C3⋊C8—21no—
316C3⋊C8—21no—
317C24—62no—
318#41—32no—
319#41—32no—
320#41—61no—
321#41—61no—
322C2×Q8—31no—
323C2×D4—34no—
324C2×D4—31no—
325C2×D4—61no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C22×C4—22no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—68no—
340D8—64no—
341D8—64no—
342M16—22no—
343M16—21no—
344M16—21no—
345M16—32no—
346M16—61no—
347M16—64no—
348M16C2×C4×D314yes—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—31no—
352C2×C8—31no—
353C2×C8—61no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8—64no—
357C2×C8C2×C4×D312yes—
358C2×C8D3×D411yescharacteristic
359C2×C8D3×D411yescharacteristic
360C2×C8#23912yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—32no—
364C4⋊C4—32no—
365C4⋊C4—122no—
366C4⋊C4D3×D411yescharacteristic
367C4⋊C4D3×D411yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—62no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—21no—
375C42—62no—
376C42—61no—
377C42—61no—
378C42—122no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3#52011yescharacteristic
382C22×C3D4211yescharacteristic
383C22×C3#55311yescharacteristic
384C2×D3—22no—
385C2×D3—41no—
386C2×D3—41no—
387C2×D3—44no—
388C2×D3—82no—
389C2×D3—82no—
390C2×D3—84no—
391C2×D3—162no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—82no—
395C3×C4C2×C4×D411yescharacteristic
396C3×C4C2×C4×D411yescharacteristic
397C3×C4#54711yescharacteristic
398C3×C4#54811yescharacteristic
399Dic3—22no—
400Dic3—42no—
401Dic3—42no—
402Dic3—42no—
403C23—61no—
404C23—64no—
405C23—122no—
406C23—122no—
407C23—242no—
408C23—242no—
409C23#100911yescharacteristic
410Q8—32no—
411Q8—61no—
412D4—32no—
413D4—62no—
414D4—62no—
415D4—61no—
416D4—64no—
417D4—64no—
418D4—122no—
419D4—122no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—82no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—482no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C2×D3×D411yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C2×C4C2×#23911yescharacteristic
439C2×C4#100711yescharacteristic
440C2×C4D3×#4111yescharacteristic
441C8—24no—
442C8—41no—
443C8—61no—
444C8—61no—
445C8—62no—
446C8#89312yes—
447C8#89312yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3#236111yescharacteristic
451C2×C3#329011yescharacteristic
452C2×C3#329211yescharacteristic
453D3—42no—
454D3—84no—
455C22—21no—
456C22—21no—
457C22—62no—
458C22—121no—
459C22—121no—
460C22—124no—
461C22—244no—
462C22—242no—
463C22—242no—
464C22—482no—
465C22#557411yescharacteristic
466C22#560911yescharacteristic
467C22#569511yescharacteristic
468C4—41no—
469C4—41no—
470C4—62no—
471C4—82no—
472C4—122no—
473C4—122no—
474C4—122no—
475C4C2×#89311yescharacteristic
476C4C4×D3×D411yescharacteristic
477C4#563311yescharacteristic
478C4#572611yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—122no—
483C2—244no—
484C2#7620611yescharacteristic
485C2#8094111yescharacteristic
486C2#8094611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1952C2
socleC22×C3D42
prime radicalC2×#4897C2
cosocleC22×C3×C4C24
hypercenter#1952D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
2. #79201C2semidirect
3. #79146C2semidirect
4. #78561C2semidirect
5. C3×#1952C2semidirect
6. #70702C2semidirect
7. #70674C2semidirect
8. #70628C2semidirect
9. #69887C2semidirect
10. #69005C2semidirect
11. #68983C2semidirect
32. #4897C4semidirect
52. #1952D3semidirect
72. #908C2×C4semidirect
77. C2×D24C2×C4semidirect
80. #901C2×C4semidirect
92. C3×#108C2×C4semidirect
100. #823C2×C4semidirect
247. #108C4×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×#48972
#792012
#791462
#785612
C3×#19522
#707022
#706742
#706282
#698872
#690052
#689832
#230463