Finite Group Databasebeta

← #10701894

Subgroups — #10701894

Order 768.

1270 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×C3×#536C211yesmaximal, characteristic
2C2×#5236C212yesmaximal
3C2×#5225C212yesmaximal, complemented
4#71592C211yesmaximal, characteristic, complemented
5#71587C211yesmaximal, characteristic, complemented
6#11007—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#5236—24no—
19#5236—24no—
20C2×#837—24no—
21#5225—24no—
22#5225—24no—
23C2×#835—24no—
24#4530—22no—
25#4526—22no—
26C2×#536D311yescharacteristic
27C2×#421—32no—
28C2×#419—32no—
29#1742—31no—
30#1740—31no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—22no—
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41D414yes—
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×D4—21no—
47C22×C3×D4C2×C411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4—21no—
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×#117—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4D412yes—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4D412yes—
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×#208D414yes—
77C22×C3⋊C8—22no—
78#837—216no—
79#835—216no—
80C22×#41C2×D311yescharacteristic
81C23×Q8C2×D311yescharacteristic
82C22×M16—32no—
83#536—24no—
84#536—24no—
85C2×C4⋊Q8Dic312yescomplemented
86C2×#124Dic312yescomplemented
87C2×#123—22no—
88C2×C22⋊Q8—22no—
89C2×#117—32no—
90C2×#117C2×D311yescharacteristic
91#421—64no—
92#421—64no—
93C2×#104—64no—
94#419—64no—
95#419—64no—
96C2×#103—64no—
97#343—62no—
98#342—62no—
99C24×C3—21no—
100C24×C3C2×D411yescharacteristic
101C3×#41—24no—
102C3×#41—24no—
103C3×#41—24no—
104C3×#41—24no—
105C3×#41—42no—
106C3×#41—42no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—24no—
112C2×C3×Q8—24no—
113C2×C3×Q8—24no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—24no—
117C2×C3×Q8—24no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C2×C3×Q8C22⋊C414yes—
128C2×C3×Q8C22⋊C412yes—
129C2×C3×D4—24no—
130C2×C3×D4—21no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—42no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4C22⋊C412yes—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—42no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—42no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C22⋊C414yes—
148C22×C3×C4C22×C411yescharacteristic
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×Dic3—42no—
157C3×C4⋊C4—216no—
158C3×C4⋊C4—24no—
159C3×C22⋊C4—216no—
160C3×C22⋊C4—24no—
161C3×C42—22no—
162C3×C42—22no—
163#217—24no—
164#217—42no—
165C4⋊Dic3—24no—
166C4⋊Dic3—42no—
167C4×Dic3—24no—
168C4×Dic3—42no—
169#208—24no—
170#208—24no—
171#208—42no—
172#208—42no—
173C2×C3⋊C8—24no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—42no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41#6714yes—
182C22×Q8—21no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2×Dic312yes—
189C22×Q8C2×Dic311yescharacteristic
190C22×Q8#6714yes—
191C22×D4—21no—
192C22×D4C2×Dic311yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4C22×D311yescharacteristic
196C2×M16—34no—
197C2×M16—62no—
198C2×M16—62no—
199C2×M16—62no—
200C2×M16—62no—
201C2×M16—62no—
202C22×C8—62no—
203C4⋊Q8—24no—
204C4⋊Q8—24no—
205#124—24no—
206#124—24no—
207#123—216no—
208C22⋊Q8—216no—
209#117—22no—
210#117—22no—
211#117—34no—
212#117—62no—
213#117—62no—
214#117—62no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4#6712yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4#6712yes—
221C2×C42—62no—
222C2×C42C2×Dic312yes—
223#104—616no—
224#103—616no—
225C23×C3—22no—
226C23×C3—42no—
227C23×C3—44no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3C2×C22⋊C411yescharacteristic
231C23×C3#12012yes—
232C23×C3#12112yes—
233C23×C3#12411yescharacteristic
234C23×C3#12711yescharacteristic
235C3×Q8—28no—
236C3×Q8—24no—
237C3×Q8—44no—
238C3×Q8—42no—
239C3×Q8—44no—
240C3×Q8—44no—
241C3×D4—24no—
242C3×D4—44no—
243C3×D4—44no—
244C3×D4—42no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4C2×C22⋊C411yescharacteristic
258C2×C3×C4C2×C22⋊C411yescharacteristic
259C2×C3×C4C4×D412yes—
260C2×C3×C4C4×D412yes—
261C2×C3×C4#12012yes—
262C2×C3×C4#12012yes—
263C2×C3×C4#12012yes—
264C2×C3×C4#12112yes—
265C2×C3×C4#12112yes—
266C2×C3×C4#12112yes—
267C2×C3×C4#12112yes—
268C2×C3×C4#12112yes—
269C2×C3×C4#12312yes—
270C2×C3×C4#12312yes—
271C2×C3×C4#12411yescharacteristic
272C2×C3×C4#12711yescharacteristic
273C2×Dic3—44no—
274C2×Dic3—84no—
275C3⋊C8—44no—
276C24—21no—
277C24C2×#6711yescharacteristic
278#41—24no—
279#41—24no—
280#41—24no—
281#41—24no—
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—41no—
304C2×Q8#21714yes—
305C2×Q8#21712yes—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4#21712yes—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—122no—
325C22×C4#21714yes—
326C22×C4D3×D412yes—
327C22×C4D3×D412yes—
328C22×C4#23712yes—
329C22×C4#23712yes—
330C22×C4C22×Dic311yescharacteristic
331C22×C4C2×#6712yes—
332C22×C4C2×#6712yes—
333C22×C4C2×#6711yescharacteristic
334M16—64no—
335M16—64no—
336M16—122no—
337M16—122no—
338C2×C8—64no—
339C2×C8—122no—
340C2×C8—122no—
341C4⋊C4—216no—
342C4⋊C4—24no—
343C4⋊C4—64no—
344C4⋊C4—122no—
345C22⋊C4—216no—
346C22⋊C4—24no—
347C22⋊C4—64no—
348C22⋊C4—122no—
349C42—22no—
350C42—22no—
351C42—64no—
352C42—122no—
353C22×C3—24no—
354C22×C3—22no—
355C22×C3—42no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—44no—
359C22×C3—42no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3#37911yescharacteristic
363C22×C3#39011yescharacteristic
364C22×C3#39011yescharacteristic
365C22×C3#39212yes—
366C22×C3#39412yes—
367C3×C4—44no—
368C3×C4—42no—
369C3×C4—44no—
370C3×C4#38612yes—
371C3×C4#38612yes—
372C3×C4#39212yes—
373C3×C4#39412yes—
374Dic3—84no—
375C23—22no—
376C23—42no—
377C23—44no—
378C23—41no—
379C23—41no—
380C23#93611yescharacteristic
381C23#93812yes—
382C23#94011yescharacteristic
383C23C2×#21711yescharacteristic
384C23#95312yes—
385Q8—28no—
386Q8—24no—
387Q8—44no—
388Q8—42no—
389Q8—44no—
390Q8—44no—
391D4—24no—
392D4—44no—
393D4—44no—
394D4—42no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—124no—
408C2×C4—244no—
409C2×C4D4×Dic312yes—
410C2×C4D4×Dic312yes—
411C2×C4#93512yes—
412C2×C4#93512yes—
413C2×C4#93611yescharacteristic
414C2×C4#93712yes—
415C2×C4#93712yes—
416C2×C4#93812yes—
417C2×C4#93912yes—
418C2×C4#93912yes—
419C2×C4#93912yes—
420C2×C4#93912yes—
421C2×C4#94011yescharacteristic
422C2×C4C2×#21711yescharacteristic
423C2×C4C2×#21711yescharacteristic
424C2×C4#95312yes—
425C8—124no—
426C2×C3—22no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—82no—
431C2×C3#143811yescharacteristic
432C2×C3#179612yes—
433C22—22no—
434C22—24no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—42no—
439C22—44no—
440C22—82no—
441C22—82no—
442C22#522111yescharacteristic
443C22#522111yescharacteristic
444C22#522212yes—
445C22#522312yes—
446C22#524711yescharacteristic
447C4—42no—
448C4—44no—
449C4—44no—
450C4—244no—
451C4#522012yes—
452C4#522012yes—
453C4#522212yes—
454C4#522312yes—
455C3#1100711yescharacteristic, complemented, Sylow
456C2—22no—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—82no—
461C2#7213111yescharacteristic
462C2#7214412yes—
subgroupG/H
centerC22#5247
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#536C2
socleC22×C3#379
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenterC2×#536D3
hypocenterC3#11007
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
3. C2×#5225C2semidirect
4. #71592C2semidirect
5. #71587C2semidirect
11. C2×C3×C4⋊Q8C4semidirect
12. C2×C3×#124C4semidirect
85. C2×C4⋊Q8Dic3semidirect
86. C2×#124Dic3semidirect
455. C3#11007semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×#52362
C2×#52252
#715922
#715872
#110073