Finite Group Databasebeta

← #10950927

Subgroups — #10950927

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5157C211yesmaximal, characteristic, complemented
2#80714C211yesmaximal, characteristic, complemented
3C3×#2184C211yesmaximal, characteristic, complemented
4#72036C211yesmaximal, characteristic, complemented
5#71469C212yesmaximal, complemented
6#71468C212yesmaximal, complemented
7#71069C211yesmaximal, characteristic, complemented
8#69201C212yesmaximal, complemented
9#69196C212yesmaximal, complemented
10D3×C22⋊C16C211yesmaximal, characteristic, complemented
11#69099C211yesmaximal, characteristic, complemented
12#33754—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×#938—21no—
15C2×#912C2211yescharacteristic, complemented
16C22×D3×C8C2211yescharacteristic
17#5606—21no—
18C2×C3×D16C2212yescomplemented
19C3×#466C2211yescharacteristic
20C2×#832—21no—
21#5157—21no—
22#5157—21no—
23#5157C2212yes—
24#5157C2212yes—
25#5157C2211yescharacteristic, complemented
26C2×#827C2212yescomplemented
27#5116C2211yescharacteristic
28C2×#818C2211yescharacteristic
29#5041—21no—
30C2×D3×C16C2212yescomplemented
31#4887C2211yescharacteristic, complemented
32#4880C2211yescharacteristic, complemented
33#4778—21no—
34#4772—21no—
35C3×#357C2212yescomplemented
36C3×C22⋊C16C2211yescharacteristic
37#4562C2212yescomplemented
38#4562C2212yescomplemented
39#4549C2211yescharacteristic
40(C2×D3)⋊C16C2212yescomplemented
41#4504C2212yescomplemented
42#4503C2212yescomplemented
43#4490C2212yescomplemented
44#3050—31no—
45C2×#466—31no—
46#2236—31no—
47#2234—32no—
48#2233—31no—
49#2217—32no—
50#2215—32no—
51#2187—31no—
52#2186—32no—
53#2184D311yescharacteristic, complemented
54C2×C22⋊C16—31no—
55C22×C3×D4—21no—
56C22×#67—41no—
57C2×#237—21no—
58C22×C4×D3D411yescharacteristic
59C2×C3×D8—21no—
60C2×C3×D8—42no—
61C2×C3×D8C2311yescharacteristic
62C2×C3×D8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C3×#121—21no—
65C2×#217—41no—
66#938—22no—
67#938—22no—
68#938—21no—
69#938—41no—
70#938—41no—
71#937—41no—
72D4×Dic3—41no—
73C2×#214—21no—
74C2×C4⋊Dic3—21no—
75#924—21no—
76C22×C3⋊C8D411yescharacteristic
77#912—42no—
78C2×Dic12C2311yescharacteristic
79C2×D3×C8—42no—
80C2×D3×C8D411yescharacteristic
81C2×D3×C8D411yescharacteristic
82C2×D3×C8C2311yescharacteristic
83C2×D3×C8C2311yescharacteristic
84#897—21no—
85#892—21no—
86#883—41no—
87#67⋊C4—41no—
88C3×D16—42no—
89C2×C3×C16C2312yes—
90C3×#107C2311yescharacteristic
91C3×#102—21no—
92#832—22no—
93#832—22no—
94#832—21no—
95#827—42no—
96#818—21no—
97#818C2311yescharacteristic
98#818C2312yes—
99#816—21no—
100C2×C3⋊C16C2312yes—
101#810—21no—
102#808—21no—
103D3×C16—42no—
104C2×#135—31no—
105C22×D8C2×D311yescharacteristic
106C23×C8—31no—
107#525—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×D16C2×D312yescomplemented
111C22×C16—32no—
112#467—31no—
113#467—31no—
114#466—32no—
115#466—32no—
116#466—31no—
117#466—61no—
118#466—61no—
119#466C2×D311yescharacteristic
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107—31no—
124#416—61no—
125C2×#102—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#357—32no—
130#357—32no—
131#357C2×D312yescomplemented
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C24×C3—41no—
136C23×D3D811yescharacteristic
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—42no—
147C2×#67—42no—
148C2×#67—84no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3D811yescharacteristic
152#237—42no—
153#237—42no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3D811yescharacteristic
157C2×C4×D3D811yescharacteristic
158C2×C4×D3C2×D411yescharacteristic
159C2×C4×D3C2×D411yescharacteristic
160C2×Dic6—21no—
161C3×D8—22no—
162C3×D8—22no—
163C3×D8—22no—
164C3×D8—42no—
165C2×C3×C8—22no—
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2411yescharacteristic
168C3×C4⋊C4—21no—
169C3×C22⋊C4—41no—
170#217—41no—
171#217—41no—
172#217—42no—
173#217—42no—
174#214—42no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177C4⋊Dic3—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—41no—
180C4⋊Dic3—41no—
181Dic3⋊C4—41no—
182C4×Dic3—41no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8C2×D411yescharacteristic
185C2×C3⋊C8C2×D411yescharacteristic
186Dic12—22no—
187D3×C8—22no—
188D3×C8—22no—
189D3×C8—41no—
190D3×C8—41no—
191C3×C16—22no—
192C3⋊C16—22no—
193C2×#41—61no—
194C22×D4—21no—
195C22×D4—121no—
196C23×C4—31no—
197#135—122no—
198C2×Q16—31no—
199C2×SD16—61no—
200C2×D8—21no—
201C2×D8—42no—
202C2×D8C22×D312yes—
203C2×D8C22×D311yescharacteristic
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8—122no—
210C22×C8C22×D311yescharacteristic
211#123—121no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214#121—21no—
215#121—62no—
216#121—62no—
217#121—61no—
218#121—121no—
219#121—121no—
220#120—121no—
221C4×D4—121no—
222C4×D4—121no—
223#117—61no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—121no—
226SD32—122no—
227D16—42no—
228C2×C16—32no—
229C2×C16—122no—
230C2×C16C22×D312yes—
231#107—31no—
232#107—32no—
233#107—61no—
234#107C22×D311yescharacteristic
235#106—61no—
236#103—61no—
237#103—61no—
238#102—21no—
239#102—62no—
240#102—62no—
241#102—61no—
242C23×C3—42no—
243C23×C3—41no—
244C23×C3—82no—
245C23×C3—82no—
246C23×C3—82no—
247C23×C3C2×D811yescharacteristic
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—41no—
251C22×D3C2×D811yescharacteristic
252C22×D3C2×D811yescharacteristic
253C3×D4—41no—
254C3×D4—42no—
255C3×D4—41no—
256C3×D4—42no—
257C3×D4—82no—
258C3×D4—82no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4C2×D811yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264#67—82no—
265#67—84no—
266#67—82no—
267#67—84no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272C2×Dic3—82no—
273C2×Dic3—82no—
274C2×Dic3C2×D811yescharacteristic
275C2×Dic3C2×D811yescharacteristic
276C4×D3—21no—
277C4×D3—21no—
278C4×D3—21no—
279C4×D3—21no—
280C4×D3—41no—
281C4×D3—41no—
282Dic6—41no—
283Dic6—41no—
284C3×C8—21no—
285C3×C8H(5,𝔽2)12yes—
286C3⋊C8—21no—
287C3⋊C8—21no—
288C24—31no—
289C24—41no—
290#41—122no—
291#41—122no—
292C2×Q8—61no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—122no—
302C2×D4—122no—
303C2×D4—244no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4D3×D411yescharacteristic
314Q16—62no—
315SD16—122no—
316D8—22no—
317D8—22no—
318D8—22no—
319D8—42no—
320C2×C8—22no—
321C2×C8—31no—
322C2×C8—31no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8—121no—
327C2×C8—121no—
328C2×C8D3×D411yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—21no—
331C4⋊C4—62no—
332C4⋊C4—61no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C22⋊C4—41no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—121no—
344C16—22no—
345C16—62no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—81no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—82no—
354C22×C3—164no—
355C22×C3#50812yes—
356C22×C3C22×D811yescharacteristic
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—44no—
363C2×D3#50912yes—
364C3×C4—21no—
365C3×C4—81no—
366C3×C4#53411yescharacteristic
367C3×C4#57511yescharacteristic
368Dic3—21no—
369Dic3—21no—
370Dic3—82no—
371Dic3—81no—
372Dic3—81no—
373C23—31no—
374C23—31no—
375C23—41no—
376C23—42no—
377C23—62no—
378C23—62no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—121no—
383C23D3×D811yescharacteristic
384Q8—121no—
385Q8—121no—
386D4—42no—
387D4—41no—
388D4—42no—
389D4—41no—
390D4—82no—
391D4—82no—
392D4—244no—
393D4—242no—
394D4—244no—
395D4—242no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—41no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—242no—
412C2×C4—242no—
413C2×C4D3×D811yescharacteristic
414C2×C4C2×D3×D411yescharacteristic
415C8—21no—
416C8—61no—
417C8—61no—
418C8#100412yes—
419C2×C3—22no—
420C2×C3—161no—
421C2×C3—162no—
422C2×C3#318611yescharacteristic
423C2×C3C2×#50811yescharacteristic
424C2×C3C2×#50911yescharacteristic
425D3—22no—
426D3—41no—
427C22—21no—
428C22—21no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—81no—
436C22—82no—
437C22—82no—
438C22—81no—
439C22—81no—
440C22—82no—
441C22—124no—
442C22—164no—
443C22#491212yes—
444C22C2×D3×D811yescharacteristic
445C4—21no—
446C4—61no—
447C4—61no—
448C4—81no—
449C4—242no—
450C4—241no—
451C4—241no—
452C4#558911yescharacteristic
453C4#577311yescharacteristic
454C3#3375411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—62no—
457C2—121no—
458C2—162no—
459C2—161no—
460C2#8000611yescharacteristic
461C2C2×#491211yescharacteristic
462C2D3×#50911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2184C2
socleC22×C3C22×D8
hypercenter#2184D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. C2×#5157C2semidirect
2. #80714C2semidirect
3. C3×#2184C2semidirect
4. #72036C2semidirect
5. #71469C2semidirect
6. #71468C2semidirect
7. #71069C2semidirect
8. #69201C2semidirect
9. #69196C2semidirect
10. D3×C22⋊C16C2semidirect
11. #69099C2semidirect
15. C2×#912C22semidirect
18. C2×C3×D16C22semidirect
25. #5157C22semidirect
26. C2×#827C22semidirect
30. C2×D3×C16C22semidirect
31. #4887C22semidirect
32. #4880C22semidirect
35. C3×#357C22semidirect
37. #4562C22semidirect
38. #4562C22semidirect
40. (C2×D3)⋊C16C22semidirect
41. #4504C22semidirect
42. #4503C22semidirect
43. #4490C22semidirect
53. #2184D3semidirect
110. C2×D16C2×D3semidirect
131. #357C2×D3semidirect
454. C3#33754semidirect

Maximal subgroups

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

subgroupindex
C2×#51572
#807142
C3×#21842
#720362
#714692
#714682
#710692
#692012
#691962
D3×C22⋊C162
#690992
#337543