Finite Group Databasebeta

← #10818110

Subgroups — #10818110

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5142C211yesmaximal, characteristic
2#83784C211yesmaximal, characteristic, complemented
3#78820C212yesmaximal, complemented
4#78561C211yesmaximal, characteristic, complemented
5C3×#1765C211yesmaximal, characteristic, complemented
6#70366C212yesmaximal, complemented
7#69893C211yesmaximal, characteristic, complemented
8#69829C211yesmaximal, characteristic, complemented
9#69825C211yesmaximal, characteristic, complemented
10#68931C212yesmaximal, complemented
11#68758C212yesmaximal, complemented
12#23048—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5827C2212yes—
15C2×#949C2212yes—
16C22×D24C2211yescharacteristic
17#5521C2212yes—
18C2×C3×#104C2211yescharacteristic
19C3×#416C2211yescharacteristic
20C3×#405C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22#5142C418yescomplemented
23#5131C2212yes—
24C2×#823C2212yes—
25#5121—22no—
26#5121C2212yes—
27C2×#821C2212yes—
28C2×#819C2211yescharacteristic
29#5106—21no—
30#5106C2211yescharacteristic
31C2×#817—21no—
32C2×#817C2211yescharacteristic
33#5101C2212yes—
34#4995C2212yes—
35#4939C2212yes—
36#4817—22no—
37#4788—22no—
38#4785—22no—
39#4756—22no—
40#4711—21no—
41#4711—21no—
42C4×D24—22no—
43C3×#341C2211yescharacteristic
44C3×#339C2211yescharacteristic
45#4554C2212yes—
46#4523C2212yes—
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2912—31no—
51#1972—31no—
52#1951—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1773—31no—
57#1765D311yescharacteristic, 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×#103D412yes—
94C3×#102D412yes—
95C3×#97—21no—
96C3×#97—22no—
97C3×C4×C8—21no—
98#823C2×C418yescomplemented
99#821—22no—
100#821—22no—
101#821—22no—
102#821—22no—
103#819C2×C414yescomplemented
104#818—22no—
105#818D412yes—
106#817—21no—
107#817—21no—
108#817—22no—
109#817D411yescharacteristic
110#817D411yescharacteristic
111#815—22no—
112C8×Dic3—22no—
113#808—22no—
114#807—22no—
115C2×AGL(1,ℤ/8)—32no—
116C22×D8—31no—
117C2×#131C2×D311yescharacteristic
118#520—32no—
119#518—32no—
120#472—38no—
121#442—61no—
122#442—61no—
123#442—62no—
124#440—62no—
125C4×SD16—62no—
126C4×D8—62no—
127C4×D8—62no—
128C2×#108—31no—
129#428—32no—
130#428—32no—
131#428—62no—
132#427—31no—
133#427—61no—
134C2×#106—31no—
135C2×#106—61no—
136C2×#104C2×D311yescharacteristic
137#418—32no—
138#416C2×D311yescharacteristic
139C2×#102—32no—
140C2×#100—32no—
141#405—32no—
142#405C2×D311yescharacteristic
143C2×#97C2×D311yescharacteristic
144#390—32no—
145#343—32no—
146#341C2×D311yescharacteristic
147#339—32no—
148#339C2×D311yescharacteristic
149C23×D3—22no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41C22×C412yes—
153C3×#41#4112yes—
154C2×C3×Q8C2×D411yescharacteristic
155C2×C3×D4—21no—
156C2×C3×D4C2×D411yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—42no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12C22×C414yes—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C3×M16—21no—
171C3×M16—24no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16C22×C412yes—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—24no—
179C2×C3×C8—21no—
180C2×C3×C8C22×C412yes—
181C2×C3×C8C2×D412yes—
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8#4112yes—
185C3×C4⋊C4C2×D412yes—
186C3×C22⋊C4—21no—
187C3×C42—21no—
188C3×C42—21no—
189C3×C42—21no—
190#217—22no—
191#217—42no—
192#215—28no—
193#213—28no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—84no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—42no—
200C4⋊Dic3C2×D412yes—
201C4⋊Dic3C2×D412yes—
202C4×Dic3—22no—
203C4×Dic3—42no—
204#208—22no—
205#208C22×C414yes—
206C2×C3⋊C8—22no—
207D24—24no—
208D24—24no—
209C2×#41C22×D311yescharacteristic
210C22×D4—32no—
211AGL(1,ℤ/8)—38no—
212AGL(1,ℤ/8)—64no—
213C2×SD16—62no—
214C2×D8—34no—
215C2×D8—62no—
216C2×D8—61no—
217C2×D8—61no—
218C2×D8—62no—
219#131—22no—
220#131C4×D314yescomplemented
221C2×M16—21no—
222C2×M16—32no—
223C2×M16C22×D311yescharacteristic
224C2×M16C22×D311yescharacteristic
225C22×C8—21no—
226C22×C8C22×D311yescharacteristic
227C22×C8C22×D311yescharacteristic
228C4×Q8—62no—
229C4×D4—62no—
230C4×D4—62no—
231C4×D4—62no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—122no—
235#117—32no—
236#117—62no—
237#117C22×D311yescharacteristic
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
240C2×C22⋊C4—122no—
241C2×C22⋊C4—122no—
242C2×C42C22×D311yescharacteristic
243#108—34no—
244#107—32no—
245#107—62no—
246#107—62no—
247#106—31no—
248#106—31no—
249#106—62no—
250#106—61no—
251#106—61no—
252#106—62no—
253#104—22no—
254#103D1212yes—
255#102—62no—
256#102—62no—
257#102—62no—
258#102—62no—
259#102D1212yes—
260#100—38no—
261#97—21no—
262#97—22no—
263#97—62no—
264C4×C8—21no—
265C4×C8—62no—
266C23×C3—21no—
267C23×C3H(5,𝔽2)11yescharacteristic
268C22×D3—24no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—82no—
272C22×D3—82no—
273C3×Q8—21no—
274C3×Q8C4×D412yes—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—21no—
278C3×D4C4×D412yes—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4C23×C411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—42no—
294C2×Dic3—82no—
295C2×Dic3—162no—
296D12—24no—
297D12—24no—
298D12—42no—
299D12—42no—
300C4×D3—162no—
301C4×D3—162no—
302C3×C8—22no—
303C3×C8—22no—
304C3×C8—41no—
305C3×C8C4×D412yes—
306C3×C8C4×D412yes—
307C3⋊C8—24no—
308C24—62no—
309#41—21no—
310#41—21no—
311#41C2×C4×D312yes—
312#41#23512yes—
313C2×Q8C2×D1211yescharacteristic
314C2×D4—21no—
315C2×D4—34no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4C2×D1211yescharacteristic
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C23×D311yescharacteristic
329SD16—68no—
330D8—68no—
331D8—64no—
332D8—64no—
333M16—21no—
334M16—24no—
335M16—21no—
336M16—21no—
337M16—34no—
338M16—62no—
339M16C2×C4×D312yes—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—24no—
344C2×C8—21no—
345C2×C8—62no—
346C2×C8C2×C4×D312yes—
347C2×C8C2×D1212yes—
348C2×C8D3×D411yescharacteristic
349C2×C8D3×D411yescharacteristic
350C2×C8#23712yes—
351C4⋊C4—32no—
352C4⋊C4—32no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—122no—
356C4⋊C4C2×D1212yes—
357C22⋊C4—21no—
358C22⋊C4—62no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—244no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—62no—
367C42—122no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—42no—
371C22×C3#52011yescharacteristic
372C22×C3D4211yescharacteristic
373C22×C3#55311yescharacteristic
374C2×D3—44no—
375C2×D3—82no—
376C2×D3—82no—
377C2×D3—84no—
378C2×D3—162no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—81no—
382C3×C4C2×C4×D411yescharacteristic
383C3×C4C2×C4×D411yescharacteristic
384C3×C4#54711yescharacteristic
385C3×C4#54811yescharacteristic
386Dic3—42no—
387Dic3—42no—
388Dic3—82no—
389C23—21no—
390C23—64no—
391C23—122no—
392C23—122no—
393C23—242no—
394C23—242no—
395C23#100911yescharacteristic
396Q8—21no—
397Q8C4×D1212yes—
398D4—22no—
399D4—22no—
400D4—21no—
401D4—64no—
402D4—64no—
403D4—122no—
404D4—122no—
405D4C4×D1212yes—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—122no—
415C2×C4—242no—
416C2×C4—482no—
417C2×C4—482no—
418C2×C4—482no—
419C2×C4C22×C4×D311yescharacteristic
420C2×C4C22×D1211yescharacteristic
421C2×C4C2×#23511yescharacteristic
422C2×C4C2×D3×D411yescharacteristic
423C2×C4C2×#23711yescharacteristic
424C2×C4#101011yescharacteristic
425C8—22no—
426C8—22no—
427C8—41no—
428C8—64no—
429C8#67⋊C412yes—
430C8#67⋊C412yes—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—42no—
434C2×C3#236111yescharacteristic
435C2×C3#328812yes—
436D3—84no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—124no—
441C22—244no—
442C22—242no—
443C22—242no—
444C22—482no—
445C22#552311yescharacteristic
446C22D4×D1211yescharacteristic
447C22#556511yescharacteristic
448C4—22no—
449C4—42no—
450C4—81no—
451C4—122no—
452C4—122no—
453C4—242no—
454C4C2×C4×D1211yescharacteristic
455C4C2×#67⋊C411yescharacteristic
456C4#555211yescharacteristic
457C4#555511yescharacteristic
458C3#2304811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—42no—
462C2—244no—
463C2#7346711yescharacteristic
464C2#7990012yes—
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC22×C3D42
prime radicalC2×#5142C2
cosocleC22×C3×C4C24
hypercenter#1765D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #83784C2semidirect
3. #78820C2semidirect
4. #78561C2semidirect
5. C3×#1765C2semidirect
6. #70366C2semidirect
7. #69893C2semidirect
8. #69829C2semidirect
9. #69825C2semidirect
10. #68931C2semidirect
11. #68758C2semidirect
22. #5142C4semidirect
57. #1765D3semidirect
61. C3×#131C2×C4semidirect
71. #949C2×C4semidirect
86. C2×D24C2×C4semidirect
98. #823C2×C4semidirect
103. #819C2×C4semidirect
220. #131C4×D3semidirect
458. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C2×#51422
#837842
#788202
#785612
C3×#17652
#703662
#698932
#698292
#698252
#689312
#687582
#230483