Finite Group Databasebeta

← #10698627

Subgroups — #10698627

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
2C3×#1990C211yesmaximal, characteristic, complemented
3C2×#4547C211yesmaximal, characteristic, complemented
4C2×#4459C211yesmaximal, characteristic
5#69584C211yesmaximal, characteristic
6#69568C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10420—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×#946—21no—
12C2×C4×D12C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×#384C2211yescharacteristic
15#4993—22no—
16#4988—21no—
17#4975—21no—
18C2×#809—21no—
19C2×#808—21no—
20C4×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×#340—22no—
24#4547—22no—
25#4529—21no—
26#4529—21no—
27#4459—22no—
28#2542—31no—
29#1990D311yescharacteristic, complemented
30#1948—31no—
31#1802—31no—
32C2×#332—31no—
33C2×D4⋊C8—31no—
34#1279—31no—
35C22×C3×Q8—21no—
36C22×D12D411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C42C2311yescharacteristic
49#946—42no—
50#946—42no—
51#946—42no—
52#946—42no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—22no—
60C2×Dic3⋊C4—22no—
61C2×C4×Dic3—21no—
62C22×C3⋊C8—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⋊#95—21no—
71#809—42no—
72#808—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C2×C4⋊Q8C2×D311yescharacteristic
78C2×C22⋊Q8—61no—
79C2×C22⋊Q8—61no—
80C2×C4×D4—31no—
81C2×#106—61no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8—31no—
84C2×#102—61no—
85C2×C4×C8C2×D311yescharacteristic
86#399—62no—
87#390—61no—
88#384—61no—
89#384C2×D311yescharacteristic
90C4×C4⋊C4—31no—
91#340—22no—
92#340—61no—
93#340—61no—
94#332—62no—
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×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
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—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—44no—
140(C2×D3)⋊C4—44no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3D812yes—
148C4⋊Dic3SD1612yes—
149Dic3⋊C4—44no—
150Dic3⋊C4—44no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—21no—
160C22×D4—31no—
161C23×C4—61no—
162C22×C8—21no—
163C22×C8—61no—
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—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4#6712yes—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192#106—122no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#102—122no—
198C4×C8—21no—
199C4×C8—21no—
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—81no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4#12411yescharacteristic
227C2×C3×C4C2×D811yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
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—42no—
254C3⋊C8—81no—
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—41no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4C2×#6711yescharacteristic
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—121no—
287C2×C8—121no—
288C2×C8—121no—
289C2×C8—121no—
290C2×C8—121no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—124no—
307C4⋊C4—124no—
308C22⋊C4—124no—
309C22⋊C4—124no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—121no—
313C42—121no—
314C42#23511yescharacteristic
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
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—161no—
336C3×C4#45911yescharacteristic
337C3×C4#46011yescharacteristic
338C3×C4#46511yescharacteristic
339C3×C4#47011yescharacteristic
340C3×C4#48311yescharacteristic
341C3×C4#48511yescharacteristic
342C3×C4#48611yescharacteristic
343C3×C4#49011yescharacteristic
344Dic3—42no—
345Dic3—42no—
346Dic3—81no—
347C23—62no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23#93711yescharacteristic
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—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4#88311yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#88511yescharacteristic
388C2×C4D3×D811yescharacteristic
389C2×C4#91111yescharacteristic
390C2×C4D3×SD1611yescharacteristic
391C2×C4D3×SD1611yescharacteristic
392C2×C4#91411yescharacteristic
393C2×C4#91511yescharacteristic
394C2×C4#91811yescharacteristic
395C2×C4#91911yescharacteristic
396C2×C4#92911yescharacteristic
397C2×C4#93911yescharacteristic
398C8—42no—
399C8—122no—
400C8—241no—
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#495911yescharacteristic
417C22#506211yescharacteristic
418C22#509511yescharacteristic
419C22#515611yescharacteristic
420C22#516911yescharacteristic
421C22#517011yescharacteristic
422C22#518711yescharacteristic
423C4—21no—
424C4—21no—
425C4—82no—
426C4—122no—
427C4—122no—
428C4—161no—
429C4—241no—
430C4#476011yescharacteristic
431C4#476211yescharacteristic
432C4#476411yescharacteristic
433C4#478611yescharacteristic
434C4#480511yescharacteristic
435C4#480711yescharacteristic
436C4#481411yescharacteristic
437C4#481611yescharacteristic
438C3#1042011yescharacteristic, complemented, Sylow
439C2—242no—
440C2#6969511yescharacteristic
441C2#7021111yescharacteristic
442C2#7027411yescharacteristic
443C2#7120511yescharacteristic
444C2#7122811yescharacteristic
445C2#7134211yescharacteristic
446C2#7136911yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1990C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1990D3
hypocenterC3#10420
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C3×#1990C2semidirect
3. C2×#4547C2semidirect
6. #69568C2semidirect
7. #67655C2semidirect
29. #1990D3semidirect
438. C3#10420semidirect

Maximal subgroups

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

subgroupindex
#832592
C3×#19902
C2×#45472
C2×#44592
#695842
#695682
#676552
#104203