Finite Group Databasebeta

← #10690944

Subgroups — #10690944

Order 768.

630 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82810C211yesmaximal, characteristic
2C2×#4650C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9152—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C22×C4×Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5229C412yescomplemented
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4978—22no—
20#4972C412yescomplemented
21#4946—21no—
22#4943—21no—
23#4650—22no—
24#4650—22no—
25C3×#328—22no—
26#4550—22no—
27#4538—22no—
28#2855—31no—
29C2×#382—31no—
30#1817—31no—
31C2×#328—31no—
32C2×#328D311yescharacteristic
33#1294—31no—
34#1294—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C23×Dic3D411yescharacteristic
40C23×Dic3D411yescharacteristic
41C22×C3×C8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×C4⋊Dic3C2×C412yes—
51C2×Dic3⋊C4—24no—
52C2×Dic3⋊C4—22no—
53C2×Dic3⋊C4—42no—
54C2×Dic3⋊C4—42no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3C2×C412yes—
60C2×C4×Dic3D412yes—
61C22×C3⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95D412yes—
66C3⋊#95—24no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×C22⋊Q8C2×D311yescharacteristic
76C22×C4⋊C4—61no—
77C22×C42—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8C2×D311yescharacteristic
80#391—32no—
81#389—32no—
82#387—62no—
83#385—61no—
84#382—62no—
85#382—62no—
86#381—61no—
87#380—31no—
88C2×#95—31no—
89C2×#95—61no—
90C2×#95C2×D311yescharacteristic
91#336—62no—
92#328—22no—
93#328—62no—
94C24×C3C2×D411yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—42no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3C22⋊C412yes—
126C22×Dic3C22⋊C412yes—
127C22×Dic3SD1612yes—
128C22×Dic3Q1612yes—
129C2×C3×C8—42no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137#217—41no—
138#217—41no—
139C4⋊Dic3—42no—
140Dic3⋊C4—44no—
141Dic3⋊C4—48no—
142C4×Dic3—22no—
143C4×Dic3—22no—
144C4×Dic3—22no—
145C4×Dic3—22no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—41no—
148C22×Q8C4×D311yescharacteristic
149C23×C4—21no—
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—61no—
153C23×C4C22×D311yescharacteristic
154C22×C8—21no—
155C22×C8—61no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4—122no—
165C2×C4⋊C4—122no—
166C2×C4⋊C4C4×D311yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—61no—
169C2×C42—32no—
170C2×C42—32no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—61no—
175C22⋊C8—21no—
176C22⋊C8—21no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#95—21no—
180#95—32no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95—64no—
185#95—61no—
186#95—61no—
187#95#6712yes—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3#12311yescharacteristic
193C23×C3C2×SD1611yescharacteristic
194C23×C3C2×Q1611yescharacteristic
195C23×C3AGL(1,ℤ/8)11yescharacteristic
196C23×C3#13711yescharacteristic
197C3×Q8—81no—
198C3×Q8—81no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—82no—
236C2×Dic3—82no—
237C2×Dic3#10314yes—
238C3×C8—81no—
239C3⋊C8—81no—
240C24D3×D411yescharacteristic
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—32no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—122no—
265C22×C4—122no—
266C22×C4—122no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C2×C4×D311yescharacteristic
270C22×C4#23511yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—122no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—122no—
284C4⋊C4—128no—
285C4⋊C4—124no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C42—62no—
291C42—62no—
292C42—62no—
293C42—62no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3#38611yescharacteristic
303C22×C3#41012yes—
304C22×C3C2×#10311yescharacteristic
305C22×C3#41711yescharacteristic
306C22×C3#42112yes—
307C22×C3#45011yescharacteristic
308C22×C3#45111yescharacteristic
309C22×C3#48311yescharacteristic
310C22×C3#48411yescharacteristic
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—82no—
316C3×C4—81no—
317C3×C4—81no—
318Dic3—24no—
319Dic3—42no—
320Dic3—82no—
321Dic3—82no—
322C23—22no—
323C23—22no—
324C23D3×C22⋊C411yescharacteristic
325C23#88311yescharacteristic
326C23D3×SD1611yescharacteristic
327C23#91411yescharacteristic
328C23D3×Q1611yescharacteristic
329C23#91811yescharacteristic
330C23#93711yescharacteristic
331Q8—81no—
332Q8—81no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—34no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—242no—
367C2×C4—242no—
368C2×C4#67⋊C411yescharacteristic
369C2×C4#88411yescharacteristic
370C2×C4C4×#6711yescharacteristic
371C2×C4#93811yescharacteristic
372C8—81no—
373C8—241no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3#151212yes—
379C2×C3#151812yes—
380C2×C3#187211yescharacteristic
381C2×C3#189711yescharacteristic
382C2×C3#189911yescharacteristic
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22#474012yes—
392C22D3×#10311yescharacteristic
393C22#480311yescharacteristic
394C22#480511yescharacteristic
395C22#481111yescharacteristic
396C22#482112yes—
397C22#495611yescharacteristic
398C22#516911yescharacteristic
399C22#518611yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—64no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—122no—
409C4—242no—
410C4—242no—
411C3#915211yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2#6831212yes—
417C2#6831412yes—
418C2#7002911yescharacteristic
419C2#7026811yescharacteristic
420C2#7038311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9152
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
14. #5229C4semidirect
20. #4972C4semidirect
411. C3#9152semidirect

Maximal subgroups

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

subgroupindex
#828102
C2×#46502
C2×C3×#3282
C2×#45382
#699102
#676752
#676122
#91523