Finite Group Databasebeta

← #10691466

Subgroups — #10691466

Order 768.

550 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2293C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic
3#71598C211yesmaximal, characteristic
4#69471C211yesmaximal, characteristic, complemented
5#69396C211yesmaximal, characteristic, complemented
6#67620C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9235—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×C3×#117—21no—
11#5470C2211yescharacteristic
12C3×#390—21no—
13C3×#389—21no—
14C3×#388—22no—
15C3×C4×C4⋊C4C2211yescharacteristic
16C3×C4×C22⋊C4—21no—
17C3×C43C2211yescharacteristic
18#5218—21no—
19#5019C412yescomplemented
20C4×#214—22no—
21#5008C412yescomplemented
22#4971—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26D4⋊(C3⋊C8)—22no—
27#4523—21no—
28#4523—21no—
29#4452—22no—
30#3101—31no—
31#2293D311yescharacteristic
32#1994—31no—
33#1503—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4C2×C412yes—
42C3×C4×D4D412yes—
43C3×#117—42no—
44C3×#117—42no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#214—22no—
58#926—21no—
59#924—21no—
60#924—21no—
61C2×#208—21no—
62#870—21no—
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#95—22no—
68C3×#95D412yes—
69#832—22no—
70#830—21no—
71#830—21no—
72#810—22no—
73#808—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×D4C2×D311yescharacteristic
82C2×#117—21no—
83#483—61no—
84#476—32no—
85#461—32no—
86#450—61no—
87C4×SD16—62no—
88#431—31no—
89#431—31no—
90#423—31no—
91#390—21no—
92#389—21no—
93#388—22no—
94C4×C4⋊C4C2×D311yescharacteristic
95C4×C22⋊C4—21no—
96C43C2×D311yescharacteristic
97#339—61no—
98#339—61no—
99#328—62no—
100D4⋊C8—62no—
101C24×C3—21no—
102C2×C3×D4—21no—
103C2×C3×D4—42no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4C22⋊C412yes—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—44no—
127C3×C22⋊C4—44no—
128C3×C42—21no—
129C3×C42—21no—
130C3×C42—21no—
131C3×C42—42no—
132C3×C42—42no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42SD1612yescomplemented
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#214—44no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×D4#6711yescharacteristic
155C23×C4—21no—
156C2×SD16—62no—
157C2×M16—61no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4C4×D312yes—
168C4×D4D1212yes—
169#117—42no—
170#117—42no—
171#117—61no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4#6711yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42#6711yescharacteristic
182C2×C42#6711yescharacteristic
183C2×C42C22×D311yescharacteristic
184#106—62no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#103—62no—
190#102—62no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193C4×C8—61no—
194C4×C8—61no—
195#95—22no—
196#95#6712yes—
197C23×C3—22no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3#12011yescharacteristic
203C3×D4—22no—
204C3×D4—22no—
205C3×D4—42no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
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—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×Dic3—41no—
235C2×Dic3—41no—
236Dic6—42no—
237Dic6—81no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240C3⋊C8—81no—
241C24—21no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—21no—
245C2×D4—42no—
246C2×D4(C2×D3)⋊C412yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C2×#6711yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—124no—
261M16—121no—
262M16—121no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4(C2×D3)⋊C412yes—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—121no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—42no—
286C42—41no—
287C42—41no—
288C42—121no—
289C42#21412yescomplemented
290C42C2×C4×D311yescharacteristic
291C42C2×D1211yescharacteristic
292C42#23511yescharacteristic
293C42#23511yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3#38611yescharacteristic
302C22×C3#45011yescharacteristic
303C22×C3#45211yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—82no—
311C3×C4#42112yes—
312C3×C4C4×SD1611yescharacteristic
313C3×C4#44011yescharacteristic
314C3×C4#46111yescharacteristic
315C3×C4#46311yescharacteristic
316Dic3—81no—
317Dic3—81no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23#95311yescharacteristic
324Q8—122no—
325Q8—241no—
326D4—22no—
327D4—22no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4C2×(C2×D3)⋊C411yescharacteristic
350C2×C4C4×#6711yescharacteristic
351C2×C4C4×#6711yescharacteristic
352C2×C4#92911yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93011yescharacteristic
355C2×C4#93211yescharacteristic
356C2×C4C2×#21411yescharacteristic
357C2×C4#95011yescharacteristic
358C2×C4#95111yescharacteristic
359C8—122no—
360C8—241no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—82no—
365C2×C3#187711yescharacteristic
366C2×C3#189711yescharacteristic
367C2×C3#190611yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—82no—
374C22—82no—
375C22#521311yescharacteristic
376C22#521811yescharacteristic
377C22#523911yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—82no—
385C4—241no—
386C4—241no—
387C4#500512yes—
388C4#501611yescharacteristic
389C4C4×#21411yescharacteristic
390C4#501811yescharacteristic
391C4#501911yescharacteristic
392C3#923511yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—82no—
396C2#7159111yescharacteristic
397C2#7161211yescharacteristic
398C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2293C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2293D3
hypocenterC3#9235
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
4. #69471C2semidirect
5. #69396C2semidirect
6. #67620C2semidirect
7. #67611C2semidirect
19. #5019C4semidirect
21. #5008C4semidirect
135. C3×C42SD16semidirect
289. C42#214semidirect
392. C3#9235semidirect

Maximal subgroups

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

subgroupindex
C3×#22932
#823142
#715982
#694712
#693962
#676202
#676112
#92353