Finite Group Databasebeta

← #10777944

Subgroups — #10777944

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2965C211yesmaximal, characteristic
2#82534C212yesmaximal
3#78475C211yesmaximal, characteristic, complemented
4#70966C212yesmaximal, complemented
5#70964C211yesmaximal, characteristic, complemented
6#70962C211yesmaximal, characteristic, complemented
7#70875C212yesmaximal, complemented
8#70873C211yesmaximal, characteristic, complemented
9#70871C211yesmaximal, characteristic, complemented
10#70777C212yesmaximal, complemented
11#69729C211yesmaximal, characteristic, complemented
12#19161—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5470C2212yes—
15C3×#446C2211yescharacteristic
16C3×#446C2211yescharacteristic
17C3×#436C2211yescharacteristic
18C3×#436C2211yescharacteristic
19C3×#435—21no—
20C3×#435—21no—
21C3×#431—21no—
22C3×#431C2211yescharacteristic
23C3×C4×M16—21no—
24C2×C3×C4×C8C2211yescharacteristic
25#5239—22no—
26#5116—21no—
27#5116—21no—
28#5031C2212yes—
29C4×#216—22no—
30#5020C2212yes—
31#5019C2212yes—
32C4×#214—22no—
33#5007C2212yes—
34#4996—22no—
35#4925C2212yes—
36#4699C2211yescharacteristic
37C4×Dic12—21no—
38C4×Dic12—21no—
39#4682C2211yescharacteristic
40#4681C2212yes—
41#4486C2212yes—
42#4486C2212yes—
43#4482C2212yes—
44#4482C2212yes—
45#4458C2212yes—
46#4457C2211yescharacteristic
47#4457C2211yescharacteristic
48#3170—31no—
49#3107—32no—
50#2965D311yescharacteristic
51#1700—32no—
52#1699—32no—
53#1694—31no—
54#1693—31no—
55#1690—31no—
56#1689—31no—
57#1564—31no—
58#1544—32no—
59C2×C3×#41D411yescharacteristic
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8D411yescharacteristic
64C3×C4×Q8D411yescharacteristic
65C3×C4×Q8C2311yescharacteristic
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4D411yescharacteristic
69C3×C4×D4C2311yescharacteristic
70C3×#117—21no—
71C3×#117D411yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#216—22no—
75C2×#214—22no—
76#926—22no—
77#924—22no—
78#924—22no—
79C2×Dic12—21no—
80C2×Dic12—21no—
81#870—22no—
82#869C2312yes—
83#869C2312yes—
84C4×Dic6—22no—
85C4×Dic6—22no—
86C3×C4⋊C8—21no—
87C3×C4⋊C8—21no—
88C3×C4⋊C8C2311yescharacteristic
89C3×C4⋊C8C2311yescharacteristic
90C3×C22⋊C8—21no—
91C3×C22⋊C8—21no—
92C3×C22⋊C8—21no—
93C3×#97—21no—
94C3×#97—21no—
95C3×C4×C8—21no—
96C3×C4×C8D411yescharacteristic
97C3×C4×C8D411yescharacteristic
98C3×C4×C8C2311yescharacteristic
99C3×C4×C8C2311yescharacteristic
100#835—22no—
101#832—22no—
102#830—22no—
103#818—21no—
104#818—21no—
105#816—22no—
106#816—22no—
107#810—22no—
108#810—22no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)C2312yes—
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8—22no—
114#533—32no—
115C4×#41C2×D311yescharacteristic
116#500—31no—
117#494—31no—
118#487—32no—
119#485—62no—
120#475—32no—
121#474—32no—
122#467—61no—
123#467—61no—
124#462—32no—
125#461—32no—
126#452—62no—
127#446C2×D311yescharacteristic
128#446C2×D311yescharacteristic
129C4×Q16—62no—
130C4×Q16—61no—
131C4×Q16—61no—
132C4×SD16—62no—
133#436C2×D311yescharacteristic
134#436C2×D311yescharacteristic
135#435—21no—
136#435—21no—
137#431—21no—
138#431—32no—
139#431C2×D311yescharacteristic
140C4×M16—21no—
141C2×C4×C8C2×D311yescharacteristic
142#335—32no—
143#334—31no—
144#334—31no—
145#332—32no—
146#332—32no—
147#332—32no—
148#332—32no—
149C3×#41—42no—
150C2×C3×Q8C2×D411yescharacteristic
151C2×C3×D4—21no—
152C2×C3×D4C2×D411yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×Dic6—22no—
160C2×Dic6—22no—
161C3×M16—42no—
162C3×M16—42no—
163C2×C3×C8—22no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—41no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4C2×D411yescharacteristic
174C3×C4⋊C4C2×D411yescharacteristic
175C3×C22⋊C4—21no—
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42C2×D411yescharacteristic
182C3×C42C2×D411yescharacteristic
183C3×C42#4111yescharacteristic
184C3×C42C2411yescharacteristic
185#217—42no—
186#216—44no—
187#214—44no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192Dic3⋊C4—42no—
193Dic3⋊C4—42no—
194C4×Dic3—42no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197Dic12—42no—
198Dic12—42no—
199C2×#41D1211yescharacteristic
200C2×Q16—62no—
201C2×Q16—61no—
202C2×Q16—61no—
203C2×SD16—62no—
204C2×M16—21no—
205C2×M16—21no—
206C22×C8#6711yescharacteristic
207C22×C8#6711yescharacteristic
208C4⋊Q8—32no—
209C4⋊Q8—32no—
210#125—62no—
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C4×Q8—62no—
214C4×Q8—62no—
215C4×Q8D1211yescharacteristic
216C4×Q8C22×D311yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4D1211yescharacteristic
220C4×D4C22×D311yescharacteristic
221#117—21no—
222#117—62no—
223#117D1211yescharacteristic
224C2×C42—21no—
225C2×C42C22×D311yescharacteristic
226#107—61no—
227#107—61no—
228#107—62no—
229#106—62no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—32no—
233C4⋊C8—32no—
234C4⋊C8C22×D311yescharacteristic
235C4⋊C8C22×D311yescharacteristic
236#103—62no—
237#103—62no—
238#103—62no—
239#103—62no—
240#103—62no—
241#102—62no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—62no—
246#97—21no—
247#97—21no—
248C4×C8—21no—
249C4×C8—62no—
250C4×C8#6711yescharacteristic
251C4×C8#6711yescharacteristic
252C4×C8C22×D311yescharacteristic
253C4×C8C22×D311yescharacteristic
254C23×C3—21no—
255C23×C3#12111yescharacteristic
256C3×Q8—22no—
257C3×D4—22no—
258C3×D4—42no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×Dic3—42no—
277C2×Dic3—42no—
278Dic6—42no—
279Dic6—42no—
280Dic6—42no—
281Dic6—42no—
282C3×C8—21no—
283C3×C8—21no—
284C3×C8—21no—
285C3×C8—41no—
286C3×C8—41no—
287C3×C8#13512yes—
288C3⋊C8—42no—
289C3⋊C8—42no—
290#41—42no—
291C2×Q8—62no—
292C2×Q8—62no—
293C2×Q8C2×D1211yescharacteristic
294C2×D4—21no—
295C2×D4C2×D1211yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C2×D1211yescharacteristic
300C22×C4#23511yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—122no—
303Q16—124no—
304Q16—122no—
305SD16—124no—
306M16—42no—
307M16—42no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8C2×#6711yescharacteristic
315C2×C8C2×#6711yescharacteristic
316C2×C8C2×#6711yescharacteristic
317C2×C8C2×#6711yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—62no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—122no—
325C4⋊C4—122no—
326C4⋊C4C2×D1211yescharacteristic
327C4⋊C4C2×D1211yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—122no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—122no—
336C42C2×D1211yescharacteristic
337C42#23511yescharacteristic
338C42C2×#6711yescharacteristic
339C42C23×D311yescharacteristic
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3C2×#12111yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4#47312yes—
354C3×C4C2×#13511yescharacteristic
355C3×C4#57512yes—
356C3×C4#57511yescharacteristic
357Dic3—82no—
358Dic3—82no—
359C23—21no—
360C23#93011yescharacteristic
361Q8—22no—
362Q8—122no—
363Q8—122no—
364Q8—122no—
365Q8—122no—
366D4—22no—
367D4—42no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4#93011yescharacteristic
382C2×C4#93011yescharacteristic
383C2×C4#93011yescharacteristic
384C2×C4C22×D1211yescharacteristic
385C2×C4C2×#23511yescharacteristic
386C2×C4C22×#6711yescharacteristic
387C8—21no—
388C8—21no—
389C8—21no—
390C8—41no—
391C8—41no—
392C8—122no—
393C8—122no—
394C8#95012yes—
395C2×C3—41no—
396C2×C3—81no—
397C2×C3#303511yescharacteristic
398C2×C3#305411yescharacteristic
399C2×C3C2×#47311yescharacteristic
400C22—21no—
401C22—21no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22C2×#93011yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—242no—
414C4—242no—
415C4#514312yes—
416C4#574812yes—
417C4C2×#95011yescharacteristic
418C4#582211yescharacteristic
419C3#1916111yescharacteristic, complemented, Sylow
420C2—41no—
421C2—81no—
422C2#8252511yescharacteristic
423C2#8377111yescharacteristic
424C2C2×#514311yescharacteristic
subgroupG/H
centerC2×C4#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2965C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenter#2965D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
3. #78475C2semidirect
4. #70966C2semidirect
5. #70964C2semidirect
6. #70962C2semidirect
7. #70875C2semidirect
8. #70873C2semidirect
9. #70871C2semidirect
10. #70777C2semidirect
11. #69729C2semidirect
419. C3#19161semidirect

Maximal subgroups

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

subgroupindex
C3×#29652
#825342
#784752
#709662
#709642
#709622
#708752
#708732
#708712
#707772
#697292
#191613