Finite Group Databasebeta

← #10818088

Subgroups — #10818088

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×#5142C211yesmaximal, characteristic
2#83785C211yesmaximal, characteristic, complemented
3#78820C212yesmaximal, complemented
4#78561C211yesmaximal, characteristic, complemented
5C3×#1764C211yesmaximal, characteristic, complemented
6#70366C212yesmaximal, complemented
7#69894C211yesmaximal, characteristic, complemented
8#69779C211yesmaximal, characteristic, complemented
9#69776C211yesmaximal, characteristic, complemented
10#68931C212yesmaximal, complemented
11#68758C212yesmaximal, complemented
12#23046—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5827C2212yes—
15C2×#949C2212yes—
16C22×D24C2211yescharacteristic
17#5521C2212yes—
18C2×C3×#104C2211yescharacteristic
19C3×#416C2211yescharacteristic
20C3×#405C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22#5142C418yescomplemented
23#5131C2212yes—
24C2×#823C2212yes—
25#5121—22no—
26#5121C2212yes—
27C2×#821C2212yes—
28C2×#819C2211yescharacteristic
29#5106—21no—
30#5106C2211yescharacteristic
31C2×#818—21no—
32C2×#818C2211yescharacteristic
33#5101C2212yes—
34#4995C2212yes—
35#4939C2212yes—
36#4817—22no—
37#4788—22no—
38#4785—22no—
39#4756—22no—
40#4711—22no—
41C4×D24—21no—
42C4×D24—21no—
43C3×#341C2211yescharacteristic
44C3×#339C2211yescharacteristic
45#4554C2212yes—
46#4523C2212yes—
47C2×#472—31no—
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#1764D311yescharacteristic, complemented
58C2×C3×#41C2311yescharacteristic
59C22×D12C2312yes—
60C3×#131—22no—
61C3×#131C2×C414yescomplemented
62C2×C3×M16—21no—
63C2×C3×M16C2311yescharacteristic
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8—21no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#117C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70#949—24no—
71#949C2×C418yescomplemented
72Q8×Dic3—22no—
73C2×#215—22no—
74D4×Dic3—22no—
75D4×Dic3—42no—
76C2×#213—22no—
77C2×(C2×D3)⋊C4—42no—
78#926—22no—
79#926C2312yes—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3C2312yes—
82C2×#208C2312yes—
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×#104—22no—
93C3×#103D411yescharacteristic
94C3×#103D411yescharacteristic
95C3×#102D411yescharacteristic
96C3×#102D411yescharacteristic
97C3×#97—21no—
98C3×C4×C8—21no—
99C3×C4×C8—22no—
100#823C2×C418yescomplemented
101#821—22no—
102#821—22no—
103#821—22no—
104#821—22no—
105#819C2×C414yescomplemented
106#818—21no—
107#818—21no—
108#818—22no—
109#818D411yescharacteristic
110#818D411yescharacteristic
111#817—22no—
112#817D412yes—
113#815—22no—
114C8×Dic3—22no—
115#808—22no—
116#807—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131C2×D311yescharacteristic
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×#108—31no—
131#428—32no—
132#428—32no—
133#428—62no—
134#427—31no—
135#427—61no—
136C2×#107—31no—
137C2×#107—61no—
138C2×#104C2×D311yescharacteristic
139#418—32no—
140#416C2×D311yescharacteristic
141C2×#102—32no—
142C2×#100—32no—
143#405—32no—
144#405C2×D311yescharacteristic
145C2×C4×C8C2×D311yescharacteristic
146#390—32no—
147#343—32no—
148#341C2×D311yescharacteristic
149#339—32no—
150#339C2×D311yescharacteristic
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—21no—
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—22no—
163C22×Dic3—42no—
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×M16—21no—
175C3×M16—21no—
176C3×M16C22×C412yes—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—24no—
181C2×C3×C8—21no—
182C2×C3×C8C22×C412yes—
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8#4112yes—
188C3×C4⋊C4C2×D411yescharacteristic
189C3×C4⋊C4C2×D411yescharacteristic
190C3×C22⋊C4—21no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—21no—
194#217—22no—
195#217—42no—
196#215—28no—
197#213—28no—
198(C2×D3)⋊C4—42no—
199(C2×D3)⋊C4—42no—
200(C2×D3)⋊C4—84no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—42no—
204C4⋊Dic3C2×D412yes—
205C4⋊Dic3C2×D412yes—
206C4×Dic3—22no—
207C4×Dic3—42no—
208#208—22no—
209#208C22×C414yes—
210C2×C3⋊C8—22no—
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—61no—
221C2×D8—61no—
222C2×D8—62no—
223#131—22no—
224#131C4×D314yescomplemented
225C2×M16—21no—
226C2×M16—32no—
227C2×M16C22×D311yescharacteristic
228C2×M16C22×D311yescharacteristic
229C22×C8—21no—
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—32no—
240#117—62no—
241#117C22×D311yescharacteristic
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42C22×D311yescharacteristic
247#108—34no—
248#107—31no—
249#107—31no—
250#107—62no—
251#107—61no—
252#107—61no—
253#107—62no—
254#106—32no—
255#106—62no—
256#106—62no—
257#104—22no—
258#103D1211yescharacteristic
259#103D1211yescharacteristic
260#102—62no—
261#102—62no—
262#102—62no—
263#102—62no—
264#102D1211yescharacteristic
265#102D1211yescharacteristic
266#100—38no—
267#97—21no—
268#97—62no—
269C4×C8—21no—
270C4×C8—22no—
271C4×C8—62no—
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—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
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—22no—
301C2×Dic3—42no—
302C2×Dic3—82no—
303C2×Dic3—162no—
304D12—24no—
305D12—24no—
306D12—42no—
307D12—42no—
308C4×D3—162no—
309C4×D3—162no—
310C3×C8—22no—
311C3×C8—21no—
312C3×C8—21no—
313C3×C8—41no—
314C3×C8C4×D412yes—
315C3×C8C4×D412yes—
316C3⋊C8—24no—
317C24—62no—
318#41—21no—
319#41—21no—
320#41C2×C4×D312yes—
321#41#23512yes—
322C2×Q8C2×D1211yescharacteristic
323C2×D4—21no—
324C2×D4—34no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4C2×D1211yescharacteristic
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—62no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—68no—
340D8—64no—
341D8—64no—
342M16—21no—
343M16—24no—
344M16—21no—
345M16—21no—
346M16—34no—
347M16—62no—
348M16C2×C4×D312yes—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—24no—
353C2×C8—21no—
354C2×C8—62no—
355C2×C8C2×C4×D312yes—
356C2×C8C2×D1211yescharacteristic
357C2×C8C2×D1211yescharacteristic
358C2×C8D3×D411yescharacteristic
359C2×C8D3×D411yescharacteristic
360C2×C8#23712yes—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4C2×D1211yescharacteristic
367C4⋊C4C2×D1211yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—62no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—62no—
378C42—122no—
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—41no—
395C3×C4—41no—
396C3×C4C2×C4×D411yescharacteristic
397C3×C4C2×C4×D411yescharacteristic
398C3×C4#54711yescharacteristic
399C3×C4#54811yescharacteristic
400Dic3—42no—
401Dic3—42no—
402Dic3—82no—
403C23—21no—
404C23—64no—
405C23—122no—
406C23—122no—
407C23—242no—
408C23—242no—
409C23#100911yescharacteristic
410Q8—21no—
411Q8C4×D1212yes—
412D4—22no—
413D4—22no—
414D4—21no—
415D4—64no—
416D4—64no—
417D4—122no—
418D4—122no—
419D4C4×D1212yes—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—122no—
431C2×C4—242no—
432C2×C4—482no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C22×D1211yescharacteristic
437C2×C4C2×#23511yescharacteristic
438C2×C4C2×D3×D411yescharacteristic
439C2×C4C2×#23711yescharacteristic
440C2×C4#101011yescharacteristic
441C8—22no—
442C8—21no—
443C8—21no—
444C8—41no—
445C8—64no—
446C8#67⋊C412yes—
447C8#67⋊C412yes—
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#552311yescharacteristic
466C22D4×D1211yescharacteristic
467C22#556511yescharacteristic
468C4—22no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—122no—
473C4—122no—
474C4—242no—
475C4C2×C4×D1211yescharacteristic
476C4C2×#67⋊C411yescharacteristic
477C4#555211yescharacteristic
478C4#555511yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—42no—
483C2—244no—
484C2#7346711yescharacteristic
485C2#7989111yescharacteristic
486C2#7989411yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1764C2
socleC22×C3D42
prime radicalC2×#5142C2
cosocleC22×C3×C4C24
hypercenter#1764D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. #83785C2semidirect
3. #78820C2semidirect
4. #78561C2semidirect
5. C3×#1764C2semidirect
6. #70366C2semidirect
7. #69894C2semidirect
8. #69779C2semidirect
9. #69776C2semidirect
10. #68931C2semidirect
11. #68758C2semidirect
22. #5142C4semidirect
57. #1764D3semidirect
61. C3×#131C2×C4semidirect
71. #949C2×C4semidirect
86. C2×D24C2×C4semidirect
100. #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
C2×#51422
#837852
#788202
#785612
C3×#17642
#703662
#698942
#697792
#697762
#689312
#687582
#230463