Finite Group Databasebeta

← #10690026

Subgroups — #10690026

Order 768.

608 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×#4841C211yesmaximal, characteristic
2#74324C211yesmaximal, characteristic
3#70460C211yesmaximal, characteristic, complemented
4C2×#4483C211yesmaximal, characteristic
5#69582C211yesmaximal, characteristic, complemented
6C3×#1296C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#8988—31nomaximal, Sylow
9C22×#204—21no—
10C2×#892—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×#384C2211yescharacteristic
16C2×#816—21no—
17#4990—22no—
18#4988—21no—
19#4987—21no—
20Dic3×C4⋊C4—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4841—22no—
24#4841C414yescomplemented
25#4551—21no—
26#4551—21no—
27#4529—22no—
28#4483—22no—
29C2×#461—31no—
30#2354—31no—
31#1990—31no—
32#1988—31no—
33C2×D4⋊C8—31no—
34#1296D311yescharacteristic, complemented
35#1279—31no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4×Dic3—22no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56C2×#204—24no—
57C2×#204—42no—
58C2×#204—42no—
59#892—42no—
60#892—42no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12C2×C412yes—
64C4×D12D412yes—
65#869—21no—
66#869—21no—
67#869—21no—
68#869C2×C412yes—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8C2×C412yes—
73C3⋊#95—22no—
74C3⋊#95—21no—
75#816—24no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C22×SD16—61no—
79C2×C4⋊Q8—31no—
80C2×C4×D4—31no—
81C2×#117—61no—
82#461—34no—
83#461—62no—
84C2×C4⋊C8C2×D311yescharacteristic
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#103—61no—
87C2×C4×C8—31no—
88#390—61no—
89#388—62no—
90#384C2×D311yescharacteristic
91#380—61no—
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94#340—61no—
95#340—61no—
96#340—62no—
97D4⋊C8—62no—
98C23×D3—21no—
99C22×C3×C4—22no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—41no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×Dic6—22no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—22no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—44no—
143(C2×D3)⋊C4—82no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3C22⋊C412yes—
153C4⋊Dic3SD1612yescomplemented
154C4×Dic3—44no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160#204—48no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—64no—
165C2×SD16—122no—
166C2×SD16—122no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—32no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174C4×D4—32no—
175C4×D4—32no—
176C4×D4—61no—
177C4×D4—61no—
178#117—122no—
179#117—122no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4D1212yes—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C42—62no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8C4×D312yes—
198#103—64no—
199C4×C8—61no—
200C4×C8—61no—
201#95—62no—
202#95—61no—
203C23×C3#12011yescharacteristic
204C22×D3—22no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4C2×C22⋊C411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×C3×C4C2×SD1611yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4#13711yescharacteristic
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242D12—22no—
243D12—22no—
244D12—41no—
245D12—41no—
246C4×D3—42no—
247C4×D3—42no—
248C4×D3—42no—
249C4×D3—42no—
250Dic6—42no—
251Dic6—42no—
252Dic6—81no—
253Dic6—81no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C24—61no—
258C2×Q8—62no—
259C2×Q8—121no—
260C2×Q8—121no—
261C2×Q8—121no—
262C2×Q8—121no—
263C2×Q8—121no—
264C2×D4—32no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C22×C4—22no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—61no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4C2×D1211yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284SD16—128no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—32no—
295C4⋊C4—32no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—62no—
310C22⋊C4—62no—
311C22⋊C4—124no—
312C22⋊C4—242no—
313C42—124no—
314C42—121no—
315C42—121no—
316C42C2×C4×D311yescharacteristic
317C42#23511yescharacteristic
318C42D3×D411yescharacteristic
319C42#23911yescharacteristic
320C22×C3#38611yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#44911yescharacteristic
323C22×C3#45011yescharacteristic
324C22×C3#45211yescharacteristic
325C22×C3#45511yescharacteristic
326C22×C3#45611yescharacteristic
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—82no—
337C3×C4#41611yescharacteristic
338C3×C4#41811yescharacteristic
339C3×C4C4×SD1611yescharacteristic
340C3×C4#44011yescharacteristic
341C3×C4#46111yescharacteristic
342C3×C4#46311yescharacteristic
343C3×C4#48211yescharacteristic
344C3×C4#48211yescharacteristic
345Dic3—42no—
346Dic3—42no—
347Dic3—82no—
348Dic3—81no—
349C23—62no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23#88211yescharacteristic
357Q8—122no—
358Q8—122no—
359Q8—241no—
360Q8—241no—
361D4—62no—
362D4—62no—
363D4—121no—
364D4—121no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4C4×D1211yescharacteristic
387C2×C4D3×C22⋊C411yescharacteristic
388C2×C4#88411yescharacteristic
389C2×C4#89311yescharacteristic
390C2×C4#89411yescharacteristic
391C2×C4#89511yescharacteristic
392C2×C4#91111yescharacteristic
393C2×C4#91111yescharacteristic
394C2×C4D3×SD1611yescharacteristic
395C2×C4#91411yescharacteristic
396C2×C4#91611yescharacteristic
397C2×C4#91611yescharacteristic
398C2×C4#91811yescharacteristic
399C2×C4#91911yescharacteristic
400C8—42no—
401C8—81no—
402C8—122no—
403C2×C3#164211yescharacteristic
404C2×C3#164611yescharacteristic
405C2×C3#165011yescharacteristic
406C2×C3#165411yescharacteristic
407C2×C3#187511yescharacteristic
408C2×C3#187711yescharacteristic
409C2×C3#188211yescharacteristic
410D3—82no—
411C22—122no—
412C22—122no—
413C22—122no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22#466511yescharacteristic
419C22#474611yescharacteristic
420C22#475211yescharacteristic
421C22#477611yescharacteristic
422C22#478111yescharacteristic
423C22#480811yescharacteristic
424C22#481011yescharacteristic
425C4—21no—
426C4—21no—
427C4—82no—
428C4—122no—
429C4—122no—
430C4—242no—
431C4—241no—
432C4#477011yescharacteristic
433C4#478711yescharacteristic
434C4#480411yescharacteristic
435C4#481811yescharacteristic
436C4#485211yescharacteristic
437C4#486611yescharacteristic
438C4#488511yescharacteristic
439C4#488911yescharacteristic
440C3#898811yescharacteristic, complemented, Sylow
441C2—242no—
442C2#6839311yescharacteristic
443C2#6843711yescharacteristic
444C2#6843911yescharacteristic
445C2#6849511yescharacteristic
446C2#6888111yescharacteristic
447C2#6889611yescharacteristic
448C2#6900411yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1296D3
hypocenterC3#8988
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #70460C2semidirect
5. #69582C2semidirect
6. C3×#1296C2semidirect
7. #67704C2semidirect
24. #4841C4semidirect
34. #1296D3semidirect
153. C4⋊Dic3SD16semidirect
440. C3#8988semidirect

Maximal subgroups

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

subgroupindex
C2×#48412
#743242
#704602
C2×#44832
#695822
C3×#12962
#677042
#89883