Finite Group Databasebeta

← #10698517

Subgroups — #10698517

Order 768.

716 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73381C211yesmaximal, characteristic
2C2×#4470C212yesmaximal, complemented
3#69932C212yesmaximal, complemented
4#69415C211yesmaximal, characteristic, complemented
5C3×#1280C211yesmaximal, characteristic, complemented
6#10403—31nomaximal, Sylow
7C2×#930—21no—
8C2×#930—21no—
9C2×#930C2212yes—
10C22×C4⋊Dic3C2211yescharacteristic
11C2×C3×C22⋊C8C2212yes—
12C2×C3×#95C2211yescharacteristic
13C2×C3⋊#95—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#821—22no—
17#4677—22no—
18#4671—22no—
19#4551—22no—
20#4470—24no—
21#2656—31no—
22#1874—32no—
23C2×#327—32no—
24#1442—31no—
25#1280D311yescharacteristic, complemented
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C22×#67—22no—
30C23×Dic3—21no—
31C23×Dic3—21no—
32C22×D12—21no—
33C22×D12—21no—
34C22×D12D412yes—
35C22×C3×C8—22no—
36#930—22no—
37#930—22no—
38#930—22no—
39#930—22no—
40#930—42no—
41#930—42no—
42#930—42no—
43#930—42no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—42no—
51C2×C4⋊Dic3D412yes—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3×#95—22no—
55C3⋊#95—21no—
56C3⋊#95—42no—
57C3⋊#95—42no—
58C3⋊#95D412yescomplemented
59#821—44no—
60C2×#121—32no—
61C2×#121—61no—
62C2×#121—61no—
63C22×C4⋊C4—31no—
64C2×#102—62no—
65C2×C22⋊C8C2×D312yes—
66#399—62no—
67#392—62no—
68C2×#95—31no—
69C2×#95—61no—
70C2×#95—61no—
71C2×#95C2×D311yescharacteristic
72#340—62no—
73#327—64no—
74C24×C3C2×D411yescharacteristic
75C23×D3—22no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4D812yes—
87C22×C3×C4SD1612yes—
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C22×C3×C4#4111yescharacteristic
92C2×#67—42no—
93C2×#67—42no—
94C2×#67—42no—
95C2×#67—42no—
96C2×#67—42no—
97C2×#67—42no—
98C22×Dic3—22no—
99C22×Dic3—22no—
100C22×Dic3—22no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C22×Dic3—41no—
108C22×Dic3—41no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—42no—
118C2×D12—42no—
119C2×D12—81no—
120C2×D12—81no—
121C2×D12—81no—
122C2×D12—81no—
123C2×C3×C8—42no—
124C2×C3×C8—42no—
125C2×C3×C8—42no—
126(C2×D3)⋊C4—44no—
127(C2×D3)⋊C4—44no—
128(C2×D3)⋊C4—42no—
129(C2×D3)⋊C4—42no—
130C4⋊Dic3—24no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—42no—
134C4⋊Dic3—42no—
135C4⋊Dic3—42no—
136C22×D4—32no—
137C22×D4—61no—
138C22×D4—61no—
139C22×D4—62no—
140C23×C4—61no—
141C23×C4—61no—
142C23×C4D1211yescharacteristic
143C23×C4D1211yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8—22no—
146#121—62no—
147#121—62no—
148#121—62no—
149#121—62no—
150#121—122no—
151#121—122no—
152#121—122no—
153#121—122no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—122no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—62no—
162#102—124no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165#95—22no—
166#95—32no—
167#95—61no—
168#95—122no—
169#95—122no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3#12011yescharacteristic
175C23×C3#12111yescharacteristic
176C23×C3#12411yescharacteristic
177C23×C3C2×D811yescharacteristic
178C23×C3C2×SD1611yescharacteristic
179C23×C3AGL(1,ℤ/8)11yescharacteristic
180C23×C3AGL(1,ℤ/8)11yescharacteristic
181C22×D3—42no—
182C22×D3—42no—
183C22×D3—42no—
184C22×D3—42no—
185C22×D3—42no—
186C22×D3—42no—
187C22×D3—42no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4#12112yes—
204C2×C3×C4#12312yes—
205#67—82no—
206#67—82no—
207#67—82no—
208#67—82no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226D12—44no—
227D12—82no—
228D12—84no—
229D12—84no—
230C3×C8—82no—
231C24—62no—
232C24C2×D1211yescharacteristic
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—122no—
236C2×D4—122no—
237C2×D4—122no—
238C2×D4—122no—
239C2×D4—122no—
240C2×D4—122no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—241no—
246C2×D4—241no—
247C2×D4—241no—
248C2×D4—241no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4#20412yes—
275C22×C4D2412yes—
276C22×C4#23512yes—
277C22×C4D3×D412yes—
278C22×C4D3×D411yescharacteristic
279C22×C4#23911yescharacteristic
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—64no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—122no—
287C4⋊C4—122no—
288C4⋊C4—122no—
289C22⋊C4—124no—
290C22⋊C4—124no—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#39411yescharacteristic
306C22×C3#44711yescharacteristic
307C22×C3#45011yescharacteristic
308C22×C3#45411yescharacteristic
309C22×C3#45511yescharacteristic
310C22×C3B(4,𝔽2)11yescharacteristic
311C22×C3#45811yescharacteristic
312C22×C3#45911yescharacteristic
313C22×C3#46011yescharacteristic
314C22×C3#48611yescharacteristic
315C22×C3#49011yescharacteristic
316C2×D3—82no—
317C2×D3—82no—
318C2×D3—82no—
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—82no—
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—42no—
326C3×C4—42no—
327Dic3—82no—
328Dic3—82no—
329Dic3—82no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—122no—
335C23—122no—
336C23—122no—
337C23—122no—
338C23—122no—
339C23—122no—
340C23—122no—
341C23#87511yescharacteristic
342C23#88211yescharacteristic
343C23#89511yescharacteristic
344C23C2×#20411yescharacteristic
345C23C2×D2411yescharacteristic
346C23#90811yescharacteristic
347C23#90811yescharacteristic
348D4—124no—
349D4—242no—
350D4—244no—
351D4—244no—
352D4—242no—
353D4—242no—
354D4—242no—
355D4—242no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4#88412yes—
389C2×C4#89412yes—
390C8—82no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#159811yescharacteristic
396C2×C3#160111yescharacteristic
397C2×C3#160711yescharacteristic
398C2×C3#161611yescharacteristic
399C2×C3#204011yescharacteristic
400C2×C3#204111yescharacteristic
401C2×C3#204411yescharacteristic
402D3—162no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22#467211yescharacteristic
423C22#469411yescharacteristic
424C22#471311yescharacteristic
425C22#473111yescharacteristic
426C22#473211yescharacteristic
427C22#474111yescharacteristic
428C22#474211yescharacteristic
429C22#474611yescharacteristic
430C22#475311yescharacteristic
431C22#484311yescharacteristic
432C22#484811yescharacteristic
433C4—22no—
434C4—22no—
435C4—42no—
436C4—42no—
437C4—242no—
438C4—242no—
439C4—242no—
440C3#1040311yescharacteristic, complemented, Sylow
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—482no—
446C2#6810811yescharacteristic
447C2#6811011yescharacteristic
448C2#6813111yescharacteristic
449C2#6813711yescharacteristic
450C2#6862911yescharacteristic
451C2#6895711yescharacteristic
452C2#6897711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3#120
prime radical#73381C2
hypercenter#1280D3
hypocenterC3#10403
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
2. C2×#4470C2semidirect
3. #69932C2semidirect
4. #69415C2semidirect
5. C3×#1280C2semidirect
25. #1280D3semidirect
58. C3⋊#95D4semidirect
440. C3#10403semidirect

Maximal subgroups

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

subgroupindex
#733812
C2×#44702
#699322
#694152
C3×#12802
#104033