Finite Group Databasebeta

← #10698628

Subgroups — #10698628

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83259C211yesmaximal, characteristic
2C2×C3×#332C211yesmaximal, characteristic, complemented
3#71787C211yesmaximal, characteristic, complemented
4#70479C211yesmaximal, characteristic
5#70454C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10420—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×#946—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#821—21no—
16C2×#817—21no—
17#4993—22no—
18#4988—21no—
19#4975—21no—
20#4935C2211yescharacteristic
21C4×C4⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#332—22no—
24#4551—21no—
25#4551—21no—
26#4529—22no—
27#4483—22no—
28#2542—31no—
29#1990—31no—
30#1948—31no—
31#1802—31no—
32C2×#332D311yescharacteristic, complemented
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×Q8—21no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42C2311yescharacteristic
47#946—42no—
48#946—42no—
49#946—42no—
50#946—42no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—21no—
63C22×C3⋊C8—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3⋊#95—21no—
73#821—42no—
74#817—42no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C2×C4⋊Q8C2×D311yescharacteristic
78C2×C22⋊Q8—61no—
79C2×C22⋊Q8—61no—
80C2×C4×D4—31no—
81C2×#106—61no—
82C2×C4⋊C8C2×D311yescharacteristic
83C2×C4⋊C8C2×D311yescharacteristic
84C2×#102—61no—
85C2×C4×C8—31no—
86#399—62no—
87#390—61no—
88#384—31no—
89#384—61no—
90C4×C4⋊C4—31no—
91#340—61no—
92#340—61no—
93#340—62no—
94#332—22no—
95D4⋊C8—62no—
96C23×D3—21no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—41no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12SD1612yes—
117C2×C4×D3—41no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—44no—
138(C2×D3)⋊C4—44no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—42no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3D812yes—
151C4⋊Dic3SD1612yes—
152Dic3⋊C4—44no—
153Dic3⋊C4—44no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—21no—
160C22×D4—31no—
161C23×C4—61no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C22⋊Q8—122no—
170C22⋊Q8—122no—
171C22⋊Q8—122no—
172C22⋊Q8—122no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4D1212yes—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192#106—122no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8#6712yes—
197#102—122no—
198C4×C8—61no—
199C4×C8—61no—
200#95—61no—
201C23×C3#12011yescharacteristic
202C22×D3—22no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×Q8—81no—
212C3×Q8—81no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×C3×C4#12311yescharacteristic
223C2×C3×C4#12411yescharacteristic
224C2×C3×C4C2×D811yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4C2×SD1611yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244D12—22no—
245D12—22no—
246D12—41no—
247D12—41no—
248C4×D3—81no—
249C4×D3—81no—
250C4×D3—81no—
251C4×D3—81no—
252C3×C8—42no—
253C3×C8—81no—
254C3⋊C8—42no—
255C24—61no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—32no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C22×C4—22no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4C2×D1211yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—122no—
302C4⋊C4—124no—
303C4⋊C4—124no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—124no—
309C22⋊C4—124no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—121no—
313C42—121no—
314C42#23511yescharacteristic
315C42#23711yescharacteristic
316C42#23711yescharacteristic
317C42C2×#6711yescharacteristic
318C22×C3#39411yescharacteristic
319C22×C3#44711yescharacteristic
320C22×C3#44811yescharacteristic
321C22×C3#45011yescharacteristic
322C22×C3#45211yescharacteristic
323C22×C3#45411yescharacteristic
324C22×C3#45611yescharacteristic
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—82no—
335C3×C4#45911yescharacteristic
336C3×C4#46011yescharacteristic
337C3×C4#46511yescharacteristic
338C3×C4#47011yescharacteristic
339C3×C4#48311yescharacteristic
340C3×C4#48511yescharacteristic
341C3×C4#48611yescharacteristic
342C3×C4#49011yescharacteristic
343Dic3—42no—
344Dic3—42no—
345Dic3—81no—
346Dic3—161no—
347C23—62no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23#88211yescharacteristic
355Q8—81no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359D4—62no—
360D4—62no—
361D4—121no—
362D4—121no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4#88311yescharacteristic
385C2×C4#88511yescharacteristic
386C2×C4#88611yescharacteristic
387C2×C4D3×D811yescharacteristic
388C2×C4#91111yescharacteristic
389C2×C4D3×SD1611yescharacteristic
390C2×C4D3×SD1611yescharacteristic
391C2×C4#91411yescharacteristic
392C2×C4#91611yescharacteristic
393C2×C4#91811yescharacteristic
394C2×C4#91811yescharacteristic
395C2×C4#93011yescharacteristic
396C2×C4#93811yescharacteristic
397C2×C4#93911yescharacteristic
398C8—42no—
399C8—81no—
400C8—122no—
401C2×C3#162711yescharacteristic
402C2×C3#163511yescharacteristic
403C2×C3#164711yescharacteristic
404C2×C3#165111yescharacteristic
405C2×C3#203511yescharacteristic
406C2×C3#203811yescharacteristic
407C2×C3#204111yescharacteristic
408D3—82no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22—241no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22#475211yescharacteristic
417C22#475311yescharacteristic
418C22#477311yescharacteristic
419C22#477611yescharacteristic
420C22#480611yescharacteristic
421C22#480811yescharacteristic
422C22#496111yescharacteristic
423C4—21no—
424C4—21no—
425C4—82no—
426C4—122no—
427C4—122no—
428C4—241no—
429C4—481no—
430C4#476011yescharacteristic
431C4#476411yescharacteristic
432C4#480511yescharacteristic
433C4#481511yescharacteristic
434C4#515011yescharacteristic
435C4#516311yescharacteristic
436C4#517111yescharacteristic
437C4#518911yescharacteristic
438C3#1042011yescharacteristic, complemented, Sylow
439C2—242no—
440C2#6843411yescharacteristic
441C2#6843611yescharacteristic
442C2#6843811yescharacteristic
443C2#6845711yescharacteristic
444C2#7010011yescharacteristic
445C2#7017711yescharacteristic
446C2#7024311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10420
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×C3×#332C2semidirect
3. #71787C2semidirect
5. #70454C2semidirect
7. #67704C2semidirect
32. C2×#332D3semidirect
438. C3#10420semidirect

Maximal subgroups

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

subgroupindex
#832592
C2×C3×#3322
#717872
#704792
#704542
C2×#44832
#677042
#104203