Finite Group Databasebeta

← #10690422

Subgroups — #10690422

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2935C211yesmaximal, characteristic
2#83290C211yesmaximal, characteristic
3#71811C211yesmaximal, characteristic
4#70429C211yesmaximal, characteristic, complemented
5#69868C211yesmaximal, characteristic, complemented
6#67704C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×#532C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#436—21no—
13C3×#435—22no—
14C3×C8×D4—21no—
15C2×C3×C4⋊C8C2211yescharacteristic
16C3×#408—21no—
17C3×C4×M16C2211yescharacteristic
18#5093C411yescharacteristic, complemented
19#5080C411yescharacteristic, complemented
20#5076C411yescharacteristic, complemented
21#5070C411yescharacteristic, complemented
22#5041—22no—
23C4×#214—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4555—21no—
28#4555—21no—
29#4537—22no—
30#4484—21no—
31#4484—21no—
32#3167—31no—
33#2935D311yescharacteristic
34#1986—31no—
35#1502—31no—
36#1481—31no—
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C3×#131—42no—
41C3×#131—42no—
42C2×C3×M16—22no—
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C22×C3×C8—22no—
46C22×C3×C8—21no—
47C3×C4⋊Q8C2×C411yescharacteristic
48C3×#127C2×C411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121D412yes—
52C3×C4×D4—21no—
53C3×C4×D4D412yes—
54C2×C3×C42C2311yescharacteristic
55C2×#214—22no—
56#926—21no—
57C2×C4⋊Dic3—21no—
58#924—21no—
59#924—21no—
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62#870—21no—
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70C3×C22⋊C8—22no—
71C3×C22⋊C8—22no—
72C3×#97—21no—
73C3×C4×C8—21no—
74#832—22no—
75C3⋊#95—21no—
76#830—21no—
77#810—22no—
78#808—22no—
79C4⋊(C3⋊C8)—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84C2×#131—21no—
85#533—31no—
86#532C2×D311yescharacteristic
87#498—31no—
88#492—31no—
89#487—31no—
90#486—31no—
91#465—62no—
92C4×SD16—62no—
93#436—21no—
94#435—22no—
95C8×D4—21no—
96#431—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98#408—21no—
99C4×M16C2×D311yescharacteristic
100C2×C4×C8—31no—
101C4×C4⋊C4—31no—
102#339—61no—
103#339—61no—
104#327—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C2×C3×Q8—21no—
112C2×C3×D4—22no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4C22⋊C412yes—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—41no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—21no—
126C3×M16—44no—
127C3×M16SD1612yescomplemented
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—44no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—41no—
144#214—44no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C2×SD16—62no—
162#131—42no—
163#131—42no—
164C2×M16—22no—
165C2×M16#6711yescharacteristic
166C2×M16#6711yescharacteristic
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8C4×D311yescharacteristic
173#127C4×D311yescharacteristic
174#125—61no—
175#124—21no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—21no—
179#121D1212yes—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×D4—21no—
183C4×D4#6712yes—
184#117—61no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8#6711yescharacteristic
192C4⋊C8#6711yescharacteristic
193#103—62no—
194#102—62no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—61no—
198#97—21no—
199C4×C8—21no—
200C4×C8—31no—
201C4×C8—31no—
202C4×C8—61no—
203C4×C8—61no—
204#95—61no—
205C23×C3—22no—
206C23×C3#12111yescharacteristic
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237Dic6—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—22no—
241C3×C8—41no—
242C3×C8—41no—
243C3×C8—41no—
244C3⋊C8—22no—
245C3⋊C8—41no—
246C3⋊C8—81no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4(C2×D3)⋊C412yes—
258C22×C4—22no—
259C22×C4—121no—
260C22×C4C2×D1211yescharacteristic
261C22×C4#23511yescharacteristic
262C22×C4C2×#6711yescharacteristic
263SD16—124no—
264M16—21no—
265M16—21no—
266M16—21no—
267M16—44no—
268M16#21412yescomplemented
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—44no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—122no—
279C2×C8—121no—
280C4⋊C4—21no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4(C2×D3)⋊C412yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—121no—
293C42—121no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3#38611yescharacteristic
306C22×C3#46111yescharacteristic
307C22×C3#46311yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4C4×SD1611yescharacteristic
313C3×C4#44011yescharacteristic
314C3×C4#44311yescharacteristic
315C3×C4#44411yescharacteristic
316C3×C4#45011yescharacteristic
317C3×C4#45211yescharacteristic
318Dic3—82no—
319Dic3—81no—
320Dic3—81no—
321C23—22no—
322C23#93011yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—122no—
326Q8—121no—
327Q8—121no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4C2×(C2×D3)⋊C411yescharacteristic
347C2×C4C4×#6711yescharacteristic
348C2×C4C4×#6711yescharacteristic
349C2×C4#92911yescharacteristic
350C2×C4#93011yescharacteristic
351C2×C4#93211yescharacteristic
352C2×C4C2×#21411yescharacteristic
353C2×C4#95011yescharacteristic
354C2×C4#95111yescharacteristic
355C2×C4#95311yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C8—62no—
361C8—121no—
362C8—241no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—82no—
366C2×C3#187611yescharacteristic
367C2×C3#187711yescharacteristic
368C2×C3#188411yescharacteristic
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22#501611yescharacteristic
375C22#501911yescharacteristic
376C22#521311yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—242no—
382C4—241no—
383C4—241no—
384C4C4×#21411yescharacteristic
385C4#501811yescharacteristic
386C4#514411yescharacteristic
387C4#514511yescharacteristic
388C4#521811yescharacteristic
389C4#523911yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—82no—
394C2#7161211yescharacteristic
395C2#7201011yescharacteristic
396C2#7202111yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2935C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#2935D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
4. #70429C2semidirect
5. #69868C2semidirect
6. #67704C2semidirect
7. #67667C2semidirect
18. #5093C4semidirect
19. #5080C4semidirect
20. #5076C4semidirect
21. #5070C4semidirect
127. C3×M16SD16semidirect
268. M16#214semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
C3×#29352
#832902
#718112
#704292
#698682
#677042
#676672
#90553