Finite Group Databasebeta

← #10698511

Subgroups — #10698511

Order 768.

760 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 115 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
2C3×#1444C211yesmaximal, characteristic, complemented
3#71560C212yesmaximal, complemented
4C2×#4451C212yesmaximal, complemented
5#67613C211yesmaximal, characteristic, complemented
6#10402—31nomaximal, Sylow
7C22×C3×C4⋊C4—21no—
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#930—21no—
10C2×#930—21no—
11C2×#930C2212yes—
12C2×C3×#95—21no—
13C2×C3×#95C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15C2×#830C2212yes—
16C2×#809—22no—
17#4677—22no—
18#4671—22no—
19#4529—22no—
20#4451—24no—
21#2656—31no—
22#1874—32no—
23C2×#327—32no—
24#1444D311yescharacteristic, complemented
25#1280—31no—
26C23×C3×C4—21no—
27C23×C3×C4—21no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—22no—
32C22×D12—21no—
33C22×D12—21no—
34C22×D12D412yes—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—42no—
37C2×C3×C4⋊C4—42no—
38C2×C3×C4⋊C4—42no—
39C2×C3×C4⋊C4—42no—
40#930—22no—
41#930—22no—
42#930—22no—
43#930—22no—
44#930—42no—
45#930—42no—
46#930—42no—
47#930—42no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54C22×C3⋊C8—22no—
55C3×#95—22no—
56C3×#95—21no—
57C3×#95—42no—
58C3×#95D412yescomplemented
59#830—22no—
60#830—22no—
61#809—44no—
62C2×#121—32no—
63C2×#121—61no—
64C2×#121—61no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×D311yescharacteristic
67C2×#102—62no—
68C2×C22⋊C8—32no—
69#399—62no—
70#392—62no—
71C2×#95—21no—
72C2×#95C2×D311yescharacteristic
73C2×#95C2×D311yescharacteristic
74#340—62no—
75#327—64no—
76C24×C3C2×D411yescharacteristic
77C23×D3—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—41no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4D812yes—
103C22×C3×C4SD1612yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4112yes—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—42no—
112C2×#67—42no—
113C2×#67—42no—
114C22×Dic3—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—42no—
118C2×D12—42no—
119C2×D12—42no—
120C2×D12—42no—
121C2×D12—81no—
122C2×D12—81no—
123C2×D12—81no—
124C2×D12—81no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4—48no—
128(C2×D3)⋊C4—44no—
129(C2×D3)⋊C4—44no—
130(C2×D3)⋊C4—42no—
131(C2×D3)⋊C4—42no—
132C4⋊Dic3—22no—
133C4⋊Dic3—22no—
134C4⋊Dic3—42no—
135C4⋊Dic3—42no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—42no—
138C2×C3⋊C8—42no—
139C22×D4—32no—
140C22×D4—61no—
141C22×D4—61no—
142C22×D4—62no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C23×C4C22×D311yescharacteristic
148C22×C8—62no—
149#121—62no—
150#121—62no—
151#121—62no—
152#121—62no—
153#121—122no—
154#121—122no—
155#121—122no—
156#121—122no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168#102—124no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171#95—21no—
172#95—22no—
173#95—42no—
174#95D1212yescomplemented
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3#12011yescharacteristic
180C23×C3#12111yescharacteristic
181C23×C3#12411yescharacteristic
182C23×C3C2×D811yescharacteristic
183C23×C3C2×SD1611yescharacteristic
184C23×C3AGL(1,ℤ/8)11yescharacteristic
185C23×C3AGL(1,ℤ/8)11yescharacteristic
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4#12112yes—
223C2×C3×C4#12312yes—
224#67—82no—
225#67—82no—
226#67—82no—
227#67—82no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231D12—44no—
232D12—82no—
233D12—84no—
234D12—84no—
235C3⋊C8—82no—
236C24—62no—
237C24C2×#6711yescharacteristic
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—122no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—122no—
246C2×D4—122no—
247C2×D4—122no—
248C2×D4—122no—
249C2×D4—122no—
250C2×D4—241no—
251C2×D4—241no—
252C2×D4—241no—
253C2×D4—241no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—62no—
279C22×C4#21312yes—
280C22×C4#21512yes—
281C22×C4C2×D1211yescharacteristic
282C22×C4#23511yescharacteristic
283C22×C4D3×D412yes—
284C22×C4#23912yes—
285C2×C8—122no—
286C2×C8—122no—
287C2×C8—122no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—48no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—122no—
294C4⋊C4—122no—
295C22⋊C4—124no—
296C22⋊C4—124no—
297C22⋊C4—122no—
298C22⋊C4—122no—
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#39411yescharacteristic
312C22×C3#44711yescharacteristic
313C22×C3#45011yescharacteristic
314C22×C3#45311yescharacteristic
315C22×C3#45611yescharacteristic
316C22×C3B(4,𝔽2)11yescharacteristic
317C22×C3#45811yescharacteristic
318C22×C3#45911yescharacteristic
319C22×C3#46011yescharacteristic
320C22×C3#48611yescharacteristic
321C22×C3#49011yescharacteristic
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—82no—
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—82no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—44no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—82no—
335Dic3—82no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—122no—
344C23—122no—
345C23—122no—
346C23—122no—
347C23#93011yescharacteristic
348C23C2×#21311yescharacteristic
349C23#93711yescharacteristic
350C23C2×#21511yescharacteristic
351C23#94711yescharacteristic
352C23#94911yescharacteristic
353C23#94911yescharacteristic
354D4—124no—
355D4—242no—
356D4—244no—
357D4—244no—
358D4—242no—
359D4—242no—
360D4—242no—
361D4—242no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4#89412yes—
395C2×C4#89512yes—
396C8—242no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3#159511yescharacteristic
402C2×C3#160411yescharacteristic
403C2×C3#161011yescharacteristic
404C2×C3#161311yescharacteristic
405C2×C3#203911yescharacteristic
406C2×C3#204111yescharacteristic
407C2×C3#204411yescharacteristic
408D3—162no—
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—242no—
422C22—242no—
423C22—242no—
424C22—242no—
425C22—242no—
426C22—242no—
427C22—242no—
428C22#499111yescharacteristic
429C22#501511yescharacteristic
430C22#502711yescharacteristic
431C22#503211yescharacteristic
432C22#503611yescharacteristic
433C22#504611yescharacteristic
434C22#505311yescharacteristic
435C22#506711yescharacteristic
436C22#507711yescharacteristic
437C22#508811yescharacteristic
438C22#509511yescharacteristic
439C4—22no—
440C4—22no—
441C4—44no—
442C4—42no—
443C4—42no—
444C4—82no—
445C4—242no—
446C3#1040211yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—482no—
452C2#6959311yescharacteristic
453C2#6959611yescharacteristic
454C2#6962711yescharacteristic
455C2#6969611yescharacteristic
456C2#6970611yescharacteristic
457C2#6971211yescharacteristic
458C2#6971411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1444C2
socleC23×C3#120
prime radical#73381C2
hypercenter#1444D3
hypocenterC3#10402
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C3×#1444C2semidirect
3. #71560C2semidirect
4. C2×#4451C2semidirect
5. #67613C2semidirect
24. #1444D3semidirect
58. C3×#95D4semidirect
174. #95D12semidirect
446. C3#10402semidirect

Maximal subgroups

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

subgroupindex
#733812
C3×#14442
#715602
C2×#44512
#676132
#104023