Finite Group Databasebeta

← #10698513

Subgroups — #10698513

Order 768.

716 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C3×#2656C211yesmaximal, characteristic
2#71677C212yesmaximal, complemented
3C2×#4537C212yesmaximal, complemented
4#69415C211yesmaximal, characteristic, complemented
5#67613C211yesmaximal, characteristic, complemented
6#10403—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×#121—21no—
9C2×C3×#121C2212yes—
10C22×C4⋊Dic3C2211yescharacteristic
11C3×#399—22no—
12C3×#392—22no—
13C2×C3×#95C2211yescharacteristic
14C2×#832—22no—
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2212yes—
19#4537—24no—
20#4529—22no—
21#2656D311yescharacteristic
22#1874—32no—
23C2×#327—32no—
24#1442—31no—
25#1280—31no—
26C22×C3×D4—22no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4D412yes—
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3—21no—
34C23×Dic3—21no—
35C3×#121—22no—
36C3×#121—22no—
37C3×#121—22no—
38C3×#121—22no—
39C3×#121—42no—
40C3×#121—42no—
41C3×#121—42no—
42C3×#121—42no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C4⋊Dic3—22no—
50C2×C4⋊Dic3—42no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53#832—44no—
54C3⋊#95—21no—
55C3⋊#95—42no—
56C3⋊#95—42no—
57C3⋊#95D412yescomplemented
58#830—22no—
59#830—22no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×D312yes—
63C22×C4⋊C4—31no—
64C2×#102—62no—
65C2×C22⋊C8—32no—
66#399—22no—
67#392—22no—
68C2×#95—31no—
69C2×#95—61no—
70C2×#95—61no—
71C2×#95C2×D311yescharacteristic
72#340—62no—
73#327—64no—
74C24×C3—22no—
75C24×C3C2×D411yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—42no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—81no—
89C2×C3×D4—81no—
90C2×C3×D4—81no—
91C2×C3×D4—81no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4D812yes—
104C22×C3×C4SD1612yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4#4112yes—
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—41no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—42no—
127C3×C22⋊C4—44no—
128C3×C22⋊C4—44no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C4⋊Dic3—24no—
132C4⋊Dic3—42no—
133C2×C3⋊C8—42no—
134C2×C3⋊C8—42no—
135C2×C3⋊C8—42no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4#6712yes—
140C23×C4—61no—
141C23×C4—61no—
142C23×C4#6711yescharacteristic
143C23×C4#6711yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8—62no—
146#121—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—42no—
151#121—42no—
152#121—42no—
153#121—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4—122no—
158C2×C4⋊C4#6712yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162#102—124no—
163C22⋊C8—62no—
164C22⋊C8—62no—
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—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—42no—
181C23×C3#12011yescharacteristic
182C23×C3#12111yescharacteristic
183C23×C3#12411yescharacteristic
184C23×C3C2×D811yescharacteristic
185C23×C3C2×SD1611yescharacteristic
186C23×C3AGL(1,ℤ/8)11yescharacteristic
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C3×D4—44no—
189C3×D4—82no—
190C3×D4—84no—
191C3×D4—84no—
192C3×D4—82no—
193C3×D4—82no—
194C3×D4—82no—
195C3×D4—82no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4#12112yes—
215C2×C3×C4#12312yes—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230C3⋊C8—82no—
231C24—22no—
232C24C2×#6711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—81no—
246C2×D4—81no—
247C2×D4—81no—
248C2×D4—81no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
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#21312yes—
275C22×C4#21412yes—
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4#23712yes—
279C22×C4C2×#6712yes—
280C2×C8—122no—
281C2×C8—122no—
282C2×C8—122no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—64no—
288C4⋊C4—122no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
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—82no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3#39411yescharacteristic
313C22×C3#44711yescharacteristic
314C22×C3#45011yescharacteristic
315C22×C3#45411yescharacteristic
316C22×C3#45511yescharacteristic
317C22×C3B(4,𝔽2)11yescharacteristic
318C22×C3#45811yescharacteristic
319C22×C3#45911yescharacteristic
320C22×C3#46011yescharacteristic
321C22×C3#48611yescharacteristic
322C22×C3#49011yescharacteristic
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4—82no—
328Dic3—82no—
329Dic3—82no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23C2×#21311yescharacteristic
342C23#93211yescharacteristic
343C23#93211yescharacteristic
344C23C2×#21411yescharacteristic
345C23#93611yescharacteristic
346C23#93811yescharacteristic
347C23#95311yescharacteristic
348D4—44no—
349D4—82no—
350D4—84no—
351D4—84no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—82no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
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#93512yes—
389C2×C4#93912yes—
390C8—242no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—162no—
396C2×C3#159811yescharacteristic
397C2×C3#160111yescharacteristic
398C2×C3#160711yescharacteristic
399C2×C3#161611yescharacteristic
400C2×C3#204011yescharacteristic
401C2×C3#204111yescharacteristic
402C2×C3#204411yescharacteristic
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—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22#507011yescharacteristic
423C22#507111yescharacteristic
424C22#507211yescharacteristic
425C22#507511yescharacteristic
426C22#521711yescharacteristic
427C22#521811yescharacteristic
428C22#522211yescharacteristic
429C22#524211yescharacteristic
430C22#524311yescharacteristic
431C22#524411yescharacteristic
432C22#524511yescharacteristic
433C4—22no—
434C4—22no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—242no—
439C4—242no—
440C3#1040311yescharacteristic, complemented, Sylow
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—162no—
446C2#7176211yescharacteristic
447C2#7184311yescharacteristic
448C2#7184611yescharacteristic
449C2#7185011yescharacteristic
450C2#7187511yescharacteristic
451C2#7187711yescharacteristic
452C2#7187811yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2656C2
socleC23×C3#120
hypercenter#2656D3
hypocenterC3#10403
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
2. #71677C2semidirect
3. C2×#4537C2semidirect
4. #69415C2semidirect
5. #67613C2semidirect
57. 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
C3×#26562
#716772
C2×#45372
#694152
#676132
#104033