Finite Group Databasebeta

← #10698569

Subgroups — #10698569

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73712C211yesmaximal, characteristic
2#70485C211yesmaximal, characteristic, complemented
3C2×#4485C211yesmaximal, characteristic
4#69772C211yesmaximal, characteristic
5C2×#4456C211yesmaximal, characteristic, complemented
6#69466C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10411—31nomaximal, Sylow
9C2×#875—21no—
10C2×#874C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#821—22no—
16C2×#816—21no—
17#4988—21no—
18#4988—21no—
19C2×#807—21no—
20#4939—21no—
21#4935C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23#4672—22no—
24#4551—21no—
25#4529—21no—
26#4485—22no—
27#4456—22no—
28#2548—31no—
29#1957—31no—
30#1947—31no—
31#1803—31no—
32C2×#331—31no—
33C2×Q8⋊C8—31no—
34#1279D311yescharacteristic, complemented
35C22×D12—21no—
36C22×D12—21no—
37C22×D12D412yes—
38C22×Dic6D411yescharacteristic
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8—21no—
56#875—42no—
57#875—42no—
58#874—21no—
59#874—21no—
60#874—21no—
61#874—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70C3×#95—21no—
71#821—44no—
72#816—42no—
73#807—42no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C2×#127—31no—
77C2×#124—61no—
78C2×C4×Q8—31no—
79C2×#107—61no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#103—61no—
83C2×#102—62no—
84C2×C4×C8C2×D311yescharacteristic
85#394—62no—
86#390—61no—
87#390—61no—
88#390—61no—
89#384—31no—
90C4×C4⋊C4C2×D311yescharacteristic
91#340—61no—
92#340—61no—
93#331—62no—
94Q8⋊C8—62no—
95C23×D3—22no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—41no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—42no—
107C2×D12—42no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6D812yes—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—44no—
143(C2×D3)⋊C4—44no—
144(C2×D3)⋊C4—82no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—42no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—31no—
161C22×D4—32no—
162C22×D4—61no—
163C22×D4—61no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—61no—
167#127—61no—
168#127—61no—
169#127—61no—
170#127—61no—
171#124—122no—
172#124—122no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4D1211yescharacteristic
183C2×C4⋊C4D1211yescharacteristic
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—62no—
188C2×C42—21no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—122no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196#103—122no—
197#102—124no—
198C4×C8—21no—
199C4×C8—21no—
200#95—21no—
201C23×C3#12011yescharacteristic
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—42no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×C3×C4#12411yescharacteristic
223C2×C3×C4C2×D811yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4#13711yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—81no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240D12—44no—
241D12—82no—
242D12—81no—
243D12—81no—
244D12—81no—
245D12—81no—
246D12—81no—
247D12—81no—
248D12—81no—
249D12—81no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C24—62no—
258C2×Q8—32no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—122no—
267C2×D4—122no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—121no—
285C22×C4C2×D1211yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—62no—
295C2×C8—121no—
296C2×C8—121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C22⋊C4—242no—
313C22⋊C4—242no—
314C42—41no—
315C42—41no—
316C42—121no—
317C42—121no—
318C42#23511yescharacteristic
319C42#23511yescharacteristic
320C42D3×D411yescharacteristic
321C42#23911yescharacteristic
322C22×C3#39411yescharacteristic
323C22×C3#44711yescharacteristic
324C22×C3#44911yescharacteristic
325C22×C3#45211yescharacteristic
326C22×C3#45211yescharacteristic
327C22×C3#45311yescharacteristic
328C22×C3#45511yescharacteristic
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—82no—
335C2×D3—82no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—81no—
339C3×C4—81no—
340C3×C4#46311yescharacteristic
341C3×C4#46411yescharacteristic
342C3×C4#46611yescharacteristic
343C3×C4#46811yescharacteristic
344C3×C4#48011yescharacteristic
345C3×C4#48211yescharacteristic
346C3×C4#48811yescharacteristic
347C3×C4#48911yescharacteristic
348Dic3—42no—
349Dic3—81no—
350Dic3—161no—
351C23—122no—
352C23—122no—
353C23—122no—
354C23—122no—
355C23—122no—
356C23—122no—
357C23—122no—
358C23#88211yescharacteristic
359Q8—62no—
360Q8—62no—
361Q8—121no—
362Q8—121no—
363D4—124no—
364D4—242no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—62no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4#87511yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4#89411yescharacteristic
394C2×C4#89411yescharacteristic
395C2×C4#89511yescharacteristic
396C2×C4#90411yescharacteristic
397C2×C4#90411yescharacteristic
398C2×C4#90811yescharacteristic
399C2×C4#90911yescharacteristic
400C2×C4D3×D811yescharacteristic
401C2×C4#91411yescharacteristic
402C2×C4#91611yescharacteristic
403C2×C4#91811yescharacteristic
404C8—42no—
405C8—81no—
406C8—122no—
407C2×C3#162111yescharacteristic
408C2×C3#163011yescharacteristic
409C2×C3#164011yescharacteristic
410C2×C3#164111yescharacteristic
411C2×C3#203311yescharacteristic
412C2×C3#203711yescharacteristic
413C2×C3#204211yescharacteristic
414D3—162no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22#467211yescharacteristic
423C22#473311yescharacteristic
424C22#473411yescharacteristic
425C22#477311yescharacteristic
426C22#480811yescharacteristic
427C22#482211yescharacteristic
428C22#482511yescharacteristic
429C4—21no—
430C4—21no—
431C4—81no—
432C4—81no—
433C4—122no—
434C4—241no—
435C4—481no—
436C4#469711yescharacteristic
437C4#471411yescharacteristic
438C4#477511yescharacteristic
439C4#481211yescharacteristic
440C4#484411yescharacteristic
441C4#484511yescharacteristic
442C4#486611yescharacteristic
443C4#488211yescharacteristic
444C3#1041111yescharacteristic, complemented, Sylow
445C2—482no—
446C2#6818611yescharacteristic
447C2#6819011yescharacteristic
448C2#6826011yescharacteristic
449C2#6826211yescharacteristic
450C2#6863011yescharacteristic
451C2#6873311yescharacteristic
452C2#6882211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#1279D3
hypocenterC3#10411
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #70485C2semidirect
5. C2×#4456C2semidirect
6. #69466C2semidirect
7. C3×#1279C2semidirect
34. #1279D3semidirect
444. C3#10411semidirect

Maximal subgroups

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

subgroupindex
#737122
#704852
C2×#44852
#697722
C2×#44562
#694662
C3×#12792
#104113