Finite Group Databasebeta

← #10701783

Subgroups — #10701783

Order 768.

1274 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×C3×#536C211yesmaximal, characteristic
2C2×#5240C211yesmaximal, characteristic
3#84074C211yesmaximal, characteristic
4C2×#5225C212yesmaximal, complemented
5C2×#4531C212yesmaximal, complemented
6#10988—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×C3×Q8C2211yescharacteristic
9C3×#536—24no—
10C3×#536—24no—
11C2×C3×C4⋊Q8C412yescomplemented
12C2×C3×#124C412yescomplemented
13C2×C3×#123—22no—
14C2×C3×C22⋊Q8—22no—
15C2×C3×#117C2211yescharacteristic
16C2×#926C2212yes—
17C22×#208C2212yes—
18#5240—24no—
19#5234—24no—
20#5225—24no—
21#5225—24no—
22C2×#835—24no—
23C2×#834—22no—
24#5207—22no—
25#4531—24no—
26C2×#536D311yescharacteristic
27C2×#419—32no—
28C2×#410—31no—
29#2879—31no—
30C2×H(3,ℤ/4)—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41—22no—
37C22×C3×Q8—21no—
38C22×C3×Q8—22no—
39C22×C3×Q8—22no—
40C22×C3×Q8—22no—
41C22×C3×Q8—22no—
42C22×C3×Q8—22no—
43C22×C3×Q8C2×C412yes—
44C22×C3×Q8C2×C411yescharacteristic
45C22×C3×Q8D414yes—
46C22×C3×D4C2×C411yescharacteristic
47C22×C3×D4D412yes—
48C23×C3×C4—21no—
49C23×C3×C4D412yes—
50C23×C3×C4C2311yescharacteristic
51C3×C4⋊Q8—24no—
52C3×C4⋊Q8—24no—
53C3×#124—24no—
54C3×#124—24no—
55C3×#123—216no—
56C3×C22⋊Q8—216no—
57C3×#117—22no—
58C3×#117D414yes—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2×C412yes—
64C2×#217—22no—
65#926—22no—
66#926—22no—
67#926—22no—
68#926D414yes—
69C2×C4⋊Dic3—22no—
70C2×C4×Dic3—22no—
71C2×#208—22no—
72C2×#208—22no—
73C2×#208—22no—
74C2×#208—22no—
75C2×#208—22no—
76C2×#208—22no—
77C22×C3⋊C8D414yes—
78#835—216no—
79#834—28no—
80#830—28no—
81C22×#41C2×D311yescharacteristic
82C23×Q8C2×D311yescharacteristic
83C22×M16—32no—
84#536—24no—
85#536—24no—
86C2×C4⋊Q8Dic312yescomplemented
87C2×#124Dic312yescomplemented
88C2×#123—22no—
89C2×C22⋊Q8—22no—
90C2×#117—32no—
91C2×#117C2×D311yescharacteristic
92#419—64no—
93#419—64no—
94C2×#103—64no—
95#410—64no—
96C2×#99—62no—
97#408—64no—
98#407—62no—
99H(3,ℤ/4)—64no—
100C24×C3C22⋊C412yes—
101C24×C3C2×D411yescharacteristic
102C3×#41—42no—
103C3×#41—42no—
104C3×#41—42no—
105C3×#41—42no—
106C3×#41—42no—
107C3×#41—42no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—24no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—41no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×Q8—41no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8C22⋊C414yes—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—41no—
146C22×C3×C4—42no—
147C22×C3×C4—41no—
148C22×C3×C4C22⋊C412yes—
149C22×C3×C4C22×C411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4112yes—
156C22×C3×C4#4112yes—
157C22×Dic3—42no—
158C3×C4⋊C4—216no—
159C3×C4⋊C4—24no—
160C3×C22⋊C4—216no—
161C3×C22⋊C4—24no—
162C3×C42—22no—
163C3×C42C22⋊C414yes—
164#217—24no—
165#217—42no—
166C4⋊Dic3—24no—
167C4⋊Dic3—42no—
168C4×Dic3—24no—
169C4×Dic3—42no—
170#208—42no—
171#208—42no—
172#208—42no—
173#208—42no—
174C2×C3⋊C8—24no—
175C2×C3⋊C8—24no—
176C2×C3⋊C8—24no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—22no—
182C2×#41—22no—
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8C2×Dic312yes—
190C22×Q8C2×Dic311yescharacteristic
191C22×Q8#6714yes—
192C22×D4C2×Dic311yescharacteristic
193C22×D4#6712yes—
194C23×C4—21no—
195C23×C4#6712yes—
196C23×C4C22×D311yescharacteristic
197C2×M16—62no—
198C2×M16—62no—
199C2×M16—62no—
200C2×M16—62no—
201C2×M16—62no—
202C2×M16—62no—
203C22×C8—34no—
204C4⋊Q8—24no—
205C4⋊Q8—24no—
206#124—24no—
207#124—24no—
208#123—216no—
209C22⋊Q8—216no—
210#117—22no—
211#117—34no—
212#117—62no—
213#117—62no—
214#117—62no—
215#117#6714yes—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—62no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—62no—
222C2×C42—62no—
223C2×C42C2×Dic312yes—
224#103—616no—
225#99—68no—
226C22⋊C8—68no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—44no—
232C23×C3C2×C22⋊C411yescharacteristic
233C23×C3#12011yescharacteristic
234C23×C3#12012yes—
235C23×C3#12112yes—
236C23×C3#12311yescharacteristic
237C3×Q8—28no—
238C3×Q8—42no—
239C3×Q8—44no—
240C3×Q8—42no—
241C3×Q8—44no—
242C3×Q8—44no—
243C3×D4—44no—
244C3×D4—44no—
245C3×D4—42no—
246C3×D4—42no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—44no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4C2×C22⋊C411yescharacteristic
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C4×D412yes—
262C2×C3×C4C4×D412yes—
263C2×C3×C4#12012yes—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12012yes—
266C2×C3×C4#12112yes—
267C2×C3×C4#12112yes—
268C2×C3×C4#12112yes—
269C2×C3×C4#12112yes—
270C2×C3×C4#12112yes—
271C2×C3×C4#12312yes—
272C2×C3×C4#12311yescharacteristic
273C2×C3×C4#12412yes—
274C2×C3×C4#12712yes—
275C2×Dic3—44no—
276C2×Dic3—84no—
277C3⋊C8—44no—
278C24#21712yes—
279C24C2×#6711yescharacteristic
280#41—42no—
281#41—42no—
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286#41—42no—
287#41—42no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8#21714yes—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—122no—
327C22×C4#21712yes—
328C22×C4D3×D412yes—
329C22×C4D3×D412yes—
330C22×C4#23712yes—
331C22×C4#23712yes—
332C22×C4C22×Dic311yescharacteristic
333C22×C4C2×#6712yes—
334C22×C4C2×#6711yescharacteristic
335C22×C4C2×#6712yes—
336M16—122no—
337M16—122no—
338M16—122no—
339M16—122no—
340C2×C8—64no—
341C2×C8—64no—
342C2×C8—64no—
343C4⋊C4—216no—
344C4⋊C4—24no—
345C4⋊C4—64no—
346C4⋊C4—122no—
347C22⋊C4—216no—
348C22⋊C4—24no—
349C22⋊C4—64no—
350C22⋊C4—122no—
351C42—22no—
352C42—64no—
353C42—122no—
354C42#21714yes—
355C22×C3—24no—
356C22×C3—22no—
357C22×C3—41no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—42no—
363C22×C3—82no—
364C22×C3#37911yescharacteristic
365C22×C3#38611yescharacteristic
366C22×C3#38611yescharacteristic
367C22×C3#39212yes—
368C22×C3#39412yes—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—44no—
372C3×C4#38612yes—
373C3×C4#39012yes—
374C3×C4#39212yes—
375C3×C4#39412yes—
376Dic3—84no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—44no—
382C23#93511yescharacteristic
383C23#93711yescharacteristic
384C23#93912yes—
385C23C2×#21711yescharacteristic
386C23#95312yes—
387Q8—28no—
388Q8—42no—
389Q8—44no—
390Q8—42no—
391Q8—44no—
392Q8—44no—
393D4—44no—
394D4—44no—
395D4—42no—
396D4—42no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—124no—
410C2×C4—244no—
411C2×C4D4×Dic312yes—
412C2×C4D4×Dic312yes—
413C2×C4#93512yes—
414C2×C4#93511yescharacteristic
415C2×C4#93612yes—
416C2×C4#93712yes—
417C2×C4#93711yescharacteristic
418C2×C4#93812yes—
419C2×C4#93812yes—
420C2×C4#93912yes—
421C2×C4#93912yes—
422C2×C4#93912yes—
423C2×C4#94012yes—
424C2×C4C2×#21711yescharacteristic
425C2×C4C2×#21711yescharacteristic
426C2×C4#95312yes—
427C8—124no—
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—82no—
433C2×C3#143811yescharacteristic
434C2×C3#190912yes—
435C22—22no—
436C22—24no—
437C22—41no—
438C22—41no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—42no—
443C22—82no—
444C22#522011yescharacteristic
445C22#522011yescharacteristic
446C22#522212yes—
447C22#522312yes—
448C22#524711yescharacteristic
449C4—44no—
450C4—42no—
451C4—44no—
452C4—244no—
453C4#522012yes—
454C4#522112yes—
455C4#522212yes—
456C4#522312yes—
457C3#1098811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—22no—
460C2—21no—
461C2—22no—
462C2—82no—
463C2#7186512yes—
464C2#7213111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#536C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#536D3
hypocenterC3#10988
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
4. C2×#5225C2semidirect
5. C2×#4531C2semidirect
11. C2×C3×C4⋊Q8C4semidirect
12. C2×C3×#124C4semidirect
86. C2×C4⋊Q8Dic3semidirect
87. C2×#124Dic3semidirect
457. C3#10988semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×#52402
#840742
C2×#52252
C2×#45312
#109883