Finite Group Databasebeta

← #10690180

Subgroups — #10690180

Order 768.

756 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73867C211yesmaximal, characteristic
2C2×#4656C211yesmaximal, characteristic
3#69963C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic
5#69525C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9014—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C22×Dic3⋊C4C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×#830C2211yescharacteristic
18C2×#816—22no—
19#4978—22no—
20C2×#810—22no—
21#4943—21no—
22Dic3×C22⋊C4—21no—
23#4656—22no—
24#4656C414yescomplemented
25#4655—22no—
26#4469—22no—
27C2×#395—31no—
28#2313—31no—
29#1787—31no—
30#1787—31no—
31C2×#324—31no—
32#1294—31no—
33#1294D311yescharacteristic
34C23×C3×C4—21no—
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×Dic6—21no—
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—44no—
45C22×Dic6C2×C412yes—
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×C4⋊Dic3—24no—
51C2×C4⋊Dic3—44no—
52C2×C4⋊Dic3D412yes—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4C2×C412yes—
58C2×Dic3⋊C4D412yes—
59C2×C4×Dic3—22no—
60C22×C3⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95C2×C412yes—
66C3⋊#95—22no—
67C3⋊#95—24no—
68C3⋊#95—21no—
69C3⋊#95—24no—
70#830—21no—
71#830—21no—
72#816—44no—
73#810—44no—
74C23×Q8—31no—
75C22×C4⋊C4—31no—
76C22×C4⋊C4—61no—
77C2×#103—62no—
78C2×#103—62no—
79C2×C22⋊C8—31no—
80C2×C22⋊C8C2×D311yescharacteristic
81#395—34no—
82#395—62no—
83#387—62no—
84#387—62no—
85#381—61no—
86#380—31no—
87#380C2×D311yescharacteristic
88C4×C22⋊C4—61no—
89C2×#95—61no—
90C2×#95C2×D311yescharacteristic
91#324—62no—
92C24×C3C2×D411yescharacteristic
93C22×C3×C4—24no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—24no—
107C22×Dic3—22no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—42no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C22×Dic3—42no—
121C22×Dic3C22⋊C412yes—
122C22×Dic3C22⋊C412yes—
123C2×Dic6—24no—
124C2×Dic6—24no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6—44no—
128C2×Dic6—44no—
129C2×Dic6—44no—
130C2×Dic6—44no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6—41no—
134C2×Dic6—41no—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140#217—41no—
141#217—41no—
142C4⋊Dic3—48no—
143C4⋊Dic3—42no—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146Dic3⋊C4—22no—
147Dic3⋊C4—22no—
148C4×Dic3—82no—
149C2×C3⋊C8—42no—
150C2×C3⋊C8—41no—
151C22×Q8—32no—
152C22×Q8—61no—
153C22×Q8—61no—
154C22×Q8—61no—
155C22×Q8—61no—
156C22×Q8—61no—
157C22×Q8—124no—
158C23×C4—21no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—61no—
162C23×C4C22×D311yescharacteristic
163C22×C8—21no—
164C22×C8—61no—
165C2×C4⋊C4—32no—
166C2×C4⋊C4—32no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—64no—
173C2×C4⋊C4—124no—
174C2×C4⋊C4D1212yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—61no—
177C2×C42—62no—
178#103—124no—
179#103—124no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#95—21no—
185#95—21no—
186#95—64no—
187#95—62no—
188#95—64no—
189#95—61no—
190#95C4×D312yes—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3C2×C22⋊C411yescharacteristic
194C23×C3#12011yescharacteristic
195C23×C3#12311yescharacteristic
196C23×C3#13511yescharacteristic
197C23×C3#13511yescharacteristic
198C23×C3#13711yescharacteristic
199C23×C3#13711yescharacteristic
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×Dic3—24no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—42no—
224C2×Dic3—44no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—44no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—81no—
237Dic6—24no—
238Dic6—24no—
239Dic6—44no—
240Dic6—44no—
241Dic6—42no—
242Dic6—42no—
243C3×C8—81no—
244C3⋊C8—81no—
245C24D3×D411yescharacteristic
246C2×Q8—64no—
247C2×Q8—64no—
248C2×Q8—62no—
249C2×Q8—62no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×Q8—124no—
253C2×Q8—124no—
254C2×Q8—124no—
255C2×Q8—124no—
256C2×Q8—121no—
257C2×Q8—121no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—64no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—62no—
278C22×C4—122no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4—122no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4C2×D1211yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23911yescharacteristic
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—122no—
291C2×C8—121no—
292C4⋊C4—42no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—122no—
298C4⋊C4—128no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—242no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3#38611yescharacteristic
313C22×C3#41611yescharacteristic
314C22×C3#41911yescharacteristic
315C22×C3#45211yescharacteristic
316C22×C3#45211yescharacteristic
317C22×C3#45612yes—
318C22×C3B(4,𝔽2)12yes—
319C22×C3#48511yescharacteristic
320C22×C3#48511yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—44no—
326C3×C4—82no—
327Dic3—42no—
328Dic3—44no—
329Dic3—42no—
330Dic3—82no—
331C23—22no—
332C23—22no—
333C23D3×C22⋊C411yescharacteristic
334C23#88211yescharacteristic
335C23#89411yescharacteristic
336C23#91211yescharacteristic
337C23#91511yescharacteristic
338C23#91611yescharacteristic
339C23#91811yescharacteristic
340Q8—64no—
341Q8—64no—
342Q8—124no—
343Q8—124no—
344Q8—122no—
345Q8—122no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—64no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—122no—
366C2×C4—124no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—124no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—241no—
379C2×C4C4×D1211yescharacteristic
380C2×C4#88411yescharacteristic
381C2×C4#89311yescharacteristic
382C2×C4#89511yescharacteristic
383C8—81no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3#160312yes—
390C2×C3#160312yes—
391C2×C3#187311yescharacteristic
392C2×C3#188611yescharacteristic
393C2×C3#188911yescharacteristic
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22#466511yescharacteristic
403C22#474112yes—
404C22#477211yescharacteristic
405C22#478311yescharacteristic
406C22#480211yescharacteristic
407C22#480811yescharacteristic
408C22#482612yes—
409C22#486811yescharacteristic
410C22#488811yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—44no—
416C4—82no—
417C4—122no—
418C4—124no—
419C4—122no—
420C4—242no—
421C3#901411yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2#6829412yes—
427C2#6832612yes—
428C2#6876511yescharacteristic
429C2#6887911yescharacteristic
430C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9014
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
24. #4656C4semidirect
421. C3#9014semidirect

Maximal subgroups

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

subgroupindex
#738672
C2×#46562
#699632
C2×#44692
#695252
C3×#12942
#676122
#90143