Finite Group Databasebeta

← #10798376

Subgroups — #10798376

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D24C211yesmaximal, characteristic
2#73761C211yesmaximal, characteristic, complemented
3#73751C211yesmaximal, characteristic, complemented
4C3×C4×#106C211yesmaximal, characteristic, complemented
5#70584C212yesmaximal, complemented
6#70569C212yesmaximal, complemented
7#69751C211yesmaximal, characteristic, complemented
8#69474C211yesmaximal, characteristic, complemented
9#69474C211yesmaximal, characteristic, complemented
10#68897C212yesmaximal
11#68814C212yesmaximal, complemented
12#21134—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C2×#893C2212yes—
15C2×C4×D12C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×#821C2211yescharacteristic
23C2×#818C2211yescharacteristic
24#4988C2212yes—
25Dic3×C4⋊C4C2212yes—
26#4972C2212yes—
27C2×#809C2212yes—
28C2×#809C2212yes—
29#4940—22no—
30C4×(C2×D3)⋊C4—22no—
31#4932C2211yescharacteristic
32C2×#803C2211yescharacteristic
33C4×D24C418yescomplemented
34#4667—21no—
35#4666—21no—
36#4664—21no—
37D3×#95—21no—
38C3×#340C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×D8—31no—
44#2319—31no—
45#2275—31no—
46#1926—32no—
47#1918—32no—
48#1879—32no—
49#1864—32no—
50C4×#106D311yescharacteristic, complemented
51#1762—31no—
52#1762—31no—
53#1746—31no—
54C22×D12C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42—21no—
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2312yes—
76C22×C3⋊C8C2312yes—
77C2×D24—24no—
78C2×D24C2×C414yes—
79#893—24no—
80#893—24no—
81C4×D12—22no—
82C4×D12—22no—
83C4×D12C2×C414yes—
84C4×D12C2×C414yescomplemented
85C3×#106—24no—
86C3×C4×C8C2×C414yescomplemented
87C3×#95—21no—
88C3×#95—21no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95D411yescharacteristic
92C3⋊#95D411yescharacteristic
93C3⋊#95D412yes—
94#821C2×C414yes—
95#821C2×C414yescomplemented
96#818C2×C414yescomplemented
97#809—24no—
98#809—24no—
99#803—22no—
100C22×D8—31no—
101C2×C4×D4—32no—
102C2×C4×D4—31no—
103C2×C4×D4—31no—
104C4×D8—38no—
105C2×#107—31no—
106C2×#106C2×D312yes—
107C2×#102—31no—
108C2×#102—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#97—31no—
112C2×C4×C8C2×D311yescharacteristic
113#390—32no—
114#389—32no—
115#388—62no—
116#381—61no—
117#380—61no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4C2×D311yescharacteristic
120C4×C4⋊C4C2×D311yescharacteristic
121C4×C22⋊C4—61no—
122C4×C22⋊C4—62no—
123#376—31no—
124C2×#95—61no—
125#340—32no—
126#340—32no—
127#340C2×D312yes—
128#336—32no—
129#336—32no—
130C23×D3—21no—
131C23×D3—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3C2×D412yes—
140C22×Dic3C2×D412yes—
141C2×D12—21no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—21no—
146C2×D12C22×C412yes—
147C2×D12C22×C412yes—
148C2×D12#4114yes—
149C2×C4×D3—28no—
150C2×C4×D3—24no—
151C2×C4×D3—24no—
152C2×C4×D3—82no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C2×C4×D3—82no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8C22×C412yes—
159C2×C3×C8C22×C412yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4#4114yes—
164C3×C42—81no—
165C3×C42—81no—
166C3×C42C22×C412yes—
167C3×C42#4112yes—
168(C2×D3)⋊C4—28no—
169(C2×D3)⋊C4—24no—
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—82no—
172(C2×D3)⋊C4—82no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3C22×C412yes—
176C4⋊Dic3C22×C412yes—
177Dic3⋊C4—82no—
178C4×Dic3—22no—
179C4×Dic3—22no—
180C4×Dic3—82no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184D24—28no—
185C22×D4—31no—
186C22×D4—31no—
187C23×C4—62no—
188C23×C4—61no—
189C23×C4—61no—
190C2×D8—34no—
191C2×D8—64no—
192C22×C8—32no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195C4×D4—34no—
196C4×D4—34no—
197C4×D4—64no—
198C4×D4—64no—
199C4×D4—62no—
200C4×D4—62no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4C22×D312yes—
205C2×C4⋊C4C22×D312yes—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—32no—
214C2×C42—62no—
215C2×C42C22×D311yescharacteristic
216#107—34no—
217#106—24no—
218#102—34no—
219#102—34no—
220#102—64no—
221#102—64no—
222#97—62no—
223C4×C8C4×D314yescomplemented
224#95—21no—
225#95—21no—
226#95—31no—
227#95—31no—
228#95—32no—
229#95—62no—
230#95—62no—
231C23×C3C22×D411yescharacteristic
232C22×D3—22no—
233C22×D3—24no—
234C22×D3—22no—
235C22×D3—42no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—41no—
241C22×D3—41no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4C4×D412yes—
252C2×C3×C4C4×D412yes—
253C2×C3×C4C23×C411yescharacteristic
254C2×C3×C4C2×#4112yes—
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4C2×#4112yes—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267D12—22no—
268D12—22no—
269D12—22no—
270D12—24no—
271D12—42no—
272D12#11714yes—
273C4×D3—84no—
274C4×D3—84no—
275C4×D3—88no—
276C3×C8—22no—
277C3×C8—22no—
278C3⋊C8—42no—
279C24—61no—
280C24—61no—
281C2×D4—32no—
282C2×D4—32no—
283C2×D4—34no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—62no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—68no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—64no—
297C22×C4—64no—
298C22×C4—242no—
299C22×C4—242no—
300C22×C4—242no—
301C22×C4—242no—
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4C23×D311yescharacteristic
305D8—68no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8C2×C4×D312yes—
312C2×C8C2×C4×D312yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—242no—
321C4⋊C4#23514yes—
322C22⋊C4—68no—
323C22⋊C4—64no—
324C22⋊C4—64no—
325C22⋊C4—242no—
326C22⋊C4—242no—
327C42—62no—
328C42—62no—
329C42—81no—
330C42—81no—
331C42—242no—
332C42C2×C4×D312yes—
333C42#23912yes—
334C22×C3C2×C4×D411yescharacteristic
335C22×C3#52512yes—
336C22×C3#57612yes—
337C22×C3#57711yescharacteristic
338C22×C3#57711yescharacteristic
339C2×D3—24no—
340C2×D3—44no—
341C2×D3—44no—
342C2×D3—42no—
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—42no—
346C2×D3—42no—
347C2×D3—82no—
348C2×D3—82no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—44no—
352C3×C4—82no—
353C3×C4C2×#11711yescharacteristic
354C3×C4C4×#4111yescharacteristic
355C3×C4#52912yes—
356Dic3—42no—
357Dic3—42no—
358Dic3—42no—
359Dic3—42no—
360C23—62no—
361C23—64no—
362C23—62no—
363C23—121no—
364C23—121no—
365C23—122no—
366C23—121no—
367C23—121no—
368C23—122no—
369C23—122no—
370C23C2×D3×D411yescharacteristic
371D4—34no—
372D4—62no—
373D4—62no—
374D4—62no—
375D4—64no—
376D4—122no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—244no—
397C2×C4—244no—
398C2×C4—248no—
399C2×C4#89312yes—
400C2×C4#89312yes—
401C2×C4C22×C4×D311yescharacteristic
402C2×C4C2×#23512yes—
403C2×C4C2×#23911yescharacteristic
404C2×C4D3×#4112yes—
405C8—22no—
406C8—22no—
407C8—122no—
408C2×C3#229011yescharacteristic
409C2×C3#294811yescharacteristic
410C2×C3#295211yescharacteristic
411C2×C3#311812yes—
412C2×C3#311912yes—
413D3—44no—
414D3—82no—
415C22—64no—
416C22—122no—
417C22—122no—
418C22—124no—
419C22—124no—
420C22—122no—
421C22—122no—
422C22—122no—
423C22—242no—
424C22—242no—
425C22C2×#89311yescharacteristic
426C22#554712yes—
427C22#576211yescharacteristic
428C22#576211yescharacteristic
429C22#576912yes—
430C4—22no—
431C4—22no—
432C4—44no—
433C4—82no—
434C4—122no—
435C4—122no—
436C4—122no—
437C4—122no—
438C4C4×#23511yescharacteristic
439C4D3×#11711yescharacteristic
440C4#558012yes—
441C3#2113411yescharacteristic, complemented, Sylow
442C2—124no—
443C2—242no—
444C2#7375411yescharacteristic
445C2#7910611yescharacteristic
446C2#7916711yescharacteristic
447C2#7969612yes—
448C2#7969712yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socleC23×C3C22×D4
prime radicalC22×D24C22
cosocleC22×C3×C4C24
hypercenterC4×#106D3
hypocenterC3#21134
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #73761C2semidirect
3. #73751C2semidirect
4. C3×C4×#106C2semidirect
5. #70584C2semidirect
6. #70569C2semidirect
7. #69751C2semidirect
8. #69474C2semidirect
9. #69474C2semidirect
11. #68814C2semidirect
33. C4×D24C4semidirect
50. C4×#106D3semidirect
84. C4×D12C2×C4semidirect
86. C3×C4×C8C2×C4semidirect
95. #821C2×C4semidirect
96. #818C2×C4semidirect
223. C4×C8C4×D3semidirect
441. C3#21134semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D242
#737612
#737512
C3×C4×#1062
#705842
#705692
#697512
#694742
#688972
#688142
#211343