Finite Group Databasebeta

← #10698508

Subgroups — #10698508

Order 768.

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

normal subgroups only

This group has 760 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#69420C211yesmaximal, characteristic, complemented
5#67613C211yesmaximal, characteristic, complemented
6#10402—31nomaximal, Sylow
7C2×C3×#121—21no—
8C2×C3×#121—21no—
9C2×C3×#121C2212yes—
10C22×C4⋊Dic3C2211yescharacteristic
11C22×Dic3⋊C4—21no—
12C3×#399—22no—
13C3×#392—22no—
14C2×C3×#95C2211yescharacteristic
15C2×#832—22no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2212yes—
19#4537—24no—
20#4529—22no—
21#2656D311yescharacteristic
22#1874—32no—
23C2×#327—32no—
24#1444—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—
51C2×Dic3⋊C4—42no—
52C2×Dic3⋊C4—42no—
53C2×Dic3⋊C4—42no—
54C22×C3⋊C8—22no—
55C3×#95—22no—
56#832—44no—
57C3⋊#95—21no—
58C3⋊#95—42no—
59C3⋊#95D412yescomplemented
60#830—22no—
61#830—22no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×D312yes—
65C22×C4⋊C4—31no—
66C22×C4⋊C4—61no—
67C2×#102—62no—
68C2×C22⋊C8—32no—
69#399—22no—
70#392—22no—
71C2×#95—31no—
72C2×#95—61no—
73C2×#95C2×D311yescharacteristic
74#340—62no—
75#327—64no—
76C24×C3—22no—
77C24×C3C2×D411yescharacteristic
78C2×C3×D4—22no—
79C2×C3×D4—22no—
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—42no—
89C2×C3×D4—42no—
90C2×C3×D4—81no—
91C2×C3×D4—81no—
92C2×C3×D4—81no—
93C2×C3×D4—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4D812yes—
106C22×C3×C4SD1612yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D412yes—
109C22×C3×C4#4112yes—
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—22no—
113C22×Dic3—22no—
114C22×Dic3—22no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—41no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—42no—
129C3×C22⋊C4—44no—
130C3×C22⋊C4—44no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—42no—
133C4⋊Dic3—24no—
134C4⋊Dic3—42no—
135Dic3⋊C4—48no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—42no—
138C2×C3⋊C8—42no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4#6712yes—
143C23×C4—61no—
144C23×C4—61no—
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C23×C4C22×D311yescharacteristic
148C22×C8—62no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—22no—
153#121—42no—
154#121—42no—
155#121—42no—
156#121—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4—122no—
161C2×C4⋊C4—122no—
162C2×C4⋊C4—122no—
163C2×C4⋊C4—122no—
164C2×C4⋊C4#6712yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168#102—124no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171#95—22no—
172#95—32no—
173#95—61no—
174#95—122no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—42no—
180C23×C3—42no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3#12011yescharacteristic
187C23×C3#12111yescharacteristic
188C23×C3#12411yescharacteristic
189C23×C3C2×D811yescharacteristic
190C23×C3C2×SD1611yescharacteristic
191C23×C3AGL(1,ℤ/8)11yescharacteristic
192C23×C3AGL(1,ℤ/8)11yescharacteristic
193C3×D4—44no—
194C3×D4—82no—
195C3×D4—84no—
196C3×D4—84no—
197C3×D4—82no—
198C3×D4—82no—
199C3×D4—82no—
200C3×D4—82no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4#12112yes—
220C2×C3×C4#12312yes—
221C2×Dic3—24no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C3⋊C8—82no—
236C24—22no—
237C24C2×#6711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—81no—
251C2×D4—81no—
252C2×D4—81no—
253C2×D4—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4#21312yes—
280C22×C4#21412yes—
281C22×C4D3×D411yescharacteristic
282C22×C4#23711yescharacteristic
283C22×C4#23712yes—
284C22×C4C2×#6712yes—
285C2×C8—122no—
286C2×C8—122no—
287C2×C8—122no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—64no—
293C4⋊C4—122no—
294C4⋊C4—128no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3#39411yescharacteristic
319C22×C3#44711yescharacteristic
320C22×C3#45011yescharacteristic
321C22×C3#45311yescharacteristic
322C22×C3#45611yescharacteristic
323C22×C3B(4,𝔽2)11yescharacteristic
324C22×C3#45811yescharacteristic
325C22×C3#45911yescharacteristic
326C22×C3#46011yescharacteristic
327C22×C3#48611yescharacteristic
328C22×C3#49011yescharacteristic
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—82no—
334Dic3—44no—
335Dic3—82no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23C2×#21311yescharacteristic
348C23#93211yescharacteristic
349C23#93211yescharacteristic
350C23C2×#21411yescharacteristic
351C23#93611yescharacteristic
352C23#93811yescharacteristic
353C23#95311yescharacteristic
354D4—44no—
355D4—82no—
356D4—84no—
357D4—84no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—64no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4#93512yes—
395C2×C4#93912yes—
396C8—242no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—162no—
402C2×C3#159511yescharacteristic
403C2×C3#160411yescharacteristic
404C2×C3#161011yescharacteristic
405C2×C3#161311yescharacteristic
406C2×C3#203911yescharacteristic
407C2×C3#204111yescharacteristic
408C2×C3#204411yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22#507011yescharacteristic
429C22#507111yescharacteristic
430C22#507211yescharacteristic
431C22#507511yescharacteristic
432C22#521711yescharacteristic
433C22#521811yescharacteristic
434C22#522211yescharacteristic
435C22#524111yescharacteristic
436C22#524311yescharacteristic
437C22#524411yescharacteristic
438C22#524611yescharacteristic
439C4—22no—
440C4—22no—
441C4—42no—
442C4—42no—
443C4—82no—
444C4—124no—
445C4—242no—
446C3#1040211yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—162no—
452C2#7176211yescharacteristic
453C2#7184411yescharacteristic
454C2#7184511yescharacteristic
455C2#7185011yescharacteristic
456C2#7187411yescharacteristic
457C2#7187611yescharacteristic
458C2#7188011yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2656C2
socleC23×C3#120
hypercenter#2656D3
hypocenterC3#10402
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. #69420C2semidirect
5. #67613C2semidirect
59. C3⋊#95D4semidirect
446. C3#10402semidirect

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
#694202
#676132
#104023