Finite Group Databasebeta

← #10818090

Subgroups — #10818090

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
1C3×#2913C211yesmaximal, characteristic, complemented
2C2×#5142C211yesmaximal, characteristic
3#82564C212yesmaximal, complemented
4#79167C211yesmaximal, characteristic, complemented
5#72008C211yesmaximal, characteristic, complemented
6#71667C212yesmaximal, complemented
7#70674C212yesmaximal, complemented
8#70673C212yesmaximal, complemented
9#70659C211yesmaximal, characteristic, complemented
10#70628C211yesmaximal, characteristic, complemented
11#70625C211yesmaximal, characteristic, complemented
12#23046—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14C3×#520C2212yes—
15C2×#949C2212yes—
16C22×D24C2211yescharacteristic
17#5521C2212yes—
18C3×#428—22no—
19C3×#428C2212yes—
20C3×#427—21no—
21C3×#427C2211yescharacteristic
22C2×C3×#107—21no—
23C2×C3×#107C2211yescharacteristic
24C2×#837C2211yescharacteristic
25#5233C2211yescharacteristic
26#5142C418yescomplemented
27C2×#823C2212yes—
28C2×#819C2211yescharacteristic
29#5101C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#5026—22no—
32C4×#215—22no—
33#5014—22no—
34C4×#213—22no—
35#5001C2212yes—
36#4997C2212yes—
37#4988C2212yes—
38C2×#809C2212yes—
39#4964C2212yes—
40#4872—21no—
41#4872—21no—
42#4869—22no—
43#4558C2211yescharacteristic
44#4555C2211yescharacteristic
45#4532C2212yes—
46#4530C2212yes—
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2913D311yescharacteristic, complemented
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×D12C2312yes—
60C3×#131—22no—
61C3×#131C2×C414yescomplemented
62C2×C3×M16—21no—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8—21no—
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4—42no—
69C3×#117—22no—
70C3×#117C2312yes—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73#949—24no—
74#949C2×C418yescomplemented
75C2×#215—22no—
76C2×#213—22no—
77C2×(C2×D3)⋊C4—42no—
78#926C2311yescharacteristic
79C2×C4×Dic3C2311yescharacteristic
80C2×#208C2311yescharacteristic
81C2×#208C2312yes—
82C22×C3⋊C8C2311yescharacteristic
83C2×D24—21no—
84C2×D24—21no—
85C2×D24—22no—
86C2×D24C2×C414yescomplemented
87#893—22no—
88#893—22no—
89D3×C22⋊C4—42no—
90C4×D12—22no—
91C4×D12—22no—
92C3×#107—21no—
93C3×#107—21no—
94C3×#107—22no—
95C3×#107D411yescharacteristic
96C3×#107D411yescharacteristic
97C3×#106—22no—
98C3×#106D412yes—
99#837—22no—
100#835D411yescharacteristic
101#835D411yescharacteristic
102#832D411yescharacteristic
103#832D411yescharacteristic
104#823C2×C418yescomplemented
105#819C2×C414yescomplemented
106#815—21no—
107C8×Dic3—21no—
108C8×Dic3—22no—
109#809—22no—
110#809—22no—
111#809—22no—
112#809—22no—
113#808—22no—
114#807—22no—
115#803—22no—
116C4×C3⋊C8—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131C2×D311yescharacteristic
120#520C2×D312yes—
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—22no—
132#428—32no—
133#428C2×D312yes—
134#427—21no—
135#427C2×D311yescharacteristic
136C2×#107—21no—
137C2×#107C2×D311yescharacteristic
138C2×#104—31no—
139#418—32no—
140#416—31no—
141C2×#102—32no—
142C2×#100—32no—
143#405—31no—
144#405—32no—
145C2×C4×C8—31no—
146#390—32no—
147#343—32no—
148#341—31no—
149#339—31no—
150#339—32no—
151C23×D3—22no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41C22×C412yes—
155C3×#41#4112yes—
156C2×C3×Q8C2×D411yescharacteristic
157C2×C3×D4—21no—
158C2×C3×D4C2×D411yescharacteristic
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—42no—
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12C22×C414yes—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C3×M16—21no—
173C3×M16—24no—
174C3×M16C22×C412yes—
175C2×C3×C8—21no—
176C2×C3×C8—24no—
177C2×C3×C8C22×C412yes—
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8#4112yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4C2×D412yes—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—42no—
188C3×C42—22no—
189C3×C42—42no—
190#217—21no—
191#215—28no—
192#213—28no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—84no—
196C4⋊Dic3C2×D411yescharacteristic
197C4⋊Dic3C2×D411yescharacteristic
198C4×Dic3—21no—
199C4×Dic3—21no—
200C4×Dic3—21no—
201#208—22no—
202#208—21no—
203#208—21no—
204#208C22×C414yes—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C2×D411yescharacteristic
210C2×C3⋊C8C2×D411yescharacteristic
211D24—24no—
212D24—24no—
213C2×#41C22×D311yescharacteristic
214C22×D4—32no—
215AGL(1,ℤ/8)—38no—
216AGL(1,ℤ/8)—64no—
217C2×SD16—62no—
218C2×D8—34no—
219C2×D8—62no—
220C2×D8—62no—
221C2×D8—61no—
222C2×D8—61no—
223#131—22no—
224#131C4×D314yescomplemented
225C2×M16—21no—
226C2×M16—32no—
227C2×M16—31no—
228C2×M16C22×D311yescharacteristic
229C22×C8—21no—
230C22×C8—31no—
231C22×C8C22×D311yescharacteristic
232C4×Q8—22no—
233C4×D4—22no—
234C4×D4—42no—
235C4×D4—62no—
236C4×D4—62no—
237C4×D4—62no—
238C4×D4—62no—
239#117—22no—
240#117—31no—
241#117C22×D312yes—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C22×D312yes—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—31no—
247#108—34no—
248#107—21no—
249#107—21no—
250#107—22no—
251#107—62no—
252#107#6711yescharacteristic
253#107#6711yescharacteristic
254#106—22no—
255#106—62no—
256#106#6712yes—
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—62no—
268#97—61no—
269C4×C8—62no—
270C4×C8—62no—
271C4×C8—61no—
272C23×C3—21no—
273C23×C3H(5,𝔽2)11yescharacteristic
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—82no—
278C22×D3—82no—
279C3×Q8—21no—
280C3×Q8C4×D412yes—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—21no—
284C3×D4C4×D412yes—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—42no—
291C2×C3×C4—82no—
292C2×C3×C4—162no—
293C2×C3×C4C23×C411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304D12—24no—
305D12—24no—
306D12—42no—
307D12—42no—
308C4×D3—162no—
309C4×D3—162no—
310C3×C8—22no—
311C3×C8C4×D412yes—
312C3×C8C4×D412yes—
313C3⋊C8—24no—
314C3⋊C8—21no—
315C3⋊C8—21no—
316C3⋊C8—41no—
317C24—62no—
318#41—21no—
319#41—21no—
320#41C2×C4×D312yes—
321#41#23912yes—
322C2×Q8D3×D411yescharacteristic
323C2×D4—21no—
324C2×D4—34no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4D3×D411yescharacteristic
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—61no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—68no—
340D8—64no—
341D8—64no—
342M16—21no—
343M16—24no—
344M16—34no—
345M16—62no—
346M16—61no—
347M16—61no—
348M16C2×C4×D312yes—
349C2×C8—21no—
350C2×C8—24no—
351C2×C8—31no—
352C2×C8—31no—
353C2×C8—61no—
354C2×C8—62no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8C2×C4×D312yes—
358C2×C8#23512yes—
359C2×C8C2×#6711yescharacteristic
360C2×C8C2×#6711yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C4⋊C4—42no—
366C4⋊C4C2×#6712yes—
367C4⋊C4C2×#6712yes—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C22⋊C4—61no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—22no—
375C42—42no—
376C42—61no—
377C42—61no—
378C42—61no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3#52011yescharacteristic
385C22×C3D4211yescharacteristic
386C22×C3#55311yescharacteristic
387C2×D3—44no—
388C2×D3—82no—
389C2×D3—82no—
390C2×D3—84no—
391C2×D3—162no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—82no—
396C3×C4C2×C4×D411yescharacteristic
397C3×C4C2×C4×D411yescharacteristic
398C3×C4#54711yescharacteristic
399C3×C4#54811yescharacteristic
400Dic3—42no—
401Dic3—41no—
402Dic3—41no—
403C23—21no—
404C23—64no—
405C23—122no—
406C23—122no—
407C23—242no—
408C23—242no—
409C23#100411yescharacteristic
410Q8—21no—
411Q8#89312yes—
412D4—22no—
413D4—22no—
414D4—21no—
415D4—64no—
416D4—64no—
417D4—122no—
418D4—122no—
419D4#89312yes—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—62no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—82no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—162no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C2×#23511yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C2×C4C2×#23911yescharacteristic
439C2×C4#100711yescharacteristic
440C2×C4C22×#6711yescharacteristic
441C8—22no—
442C8—64no—
443C8—61no—
444C8—61no—
445C8—121no—
446C8C4×#6712yes—
447C8C4×#6712yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—42no—
451C2×C3#236111yescharacteristic
452C2×C3#329011yescharacteristic
453C2×C3#329211yescharacteristic
454D3—84no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—124no—
461C22—244no—
462C22—242no—
463C22—242no—
464C22—482no—
465C22#550411yescharacteristic
466C22#551411yescharacteristic
467C22D4×#6711yescharacteristic
468C4—22no—
469C4—42no—
470C4—42no—
471C4—82no—
472C4—122no—
473C4—121no—
474C4—121no—
475C4C2×#89311yescharacteristic
476C4#550711yescharacteristic
477C4C2×C4×#6711yescharacteristic
478C4#581711yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—42no—
483C2—244no—
484C2#8216311yescharacteristic
485C2#8401011yescharacteristic
486C2#8402011yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2913C2
socleC22×C3D42
prime radicalC2×#5142C2
cosocleC22×C3×C4C24
hypercenter#2913D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. C3×#2913C2semidirect
3. #82564C2semidirect
4. #79167C2semidirect
5. #72008C2semidirect
6. #71667C2semidirect
7. #70674C2semidirect
8. #70673C2semidirect
9. #70659C2semidirect
10. #70628C2semidirect
11. #70625C2semidirect
26. #5142C4semidirect
50. #2913D3semidirect
61. C3×#131C2×C4semidirect
74. #949C2×C4semidirect
86. C2×D24C2×C4semidirect
104. #823C2×C4semidirect
105. #819C2×C4semidirect
224. #131C4×D3semidirect
479. C3#23046semidirect

Maximal subgroups

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

subgroupindex
C3×#29132
C2×#51422
#825642
#791672
#720082
#716672
#706742
#706732
#706592
#706282
#706252
#230463