Finite Group Databasebeta

← #10690415

Subgroups — #10690415

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2962C211yesmaximal, characteristic, complemented
2#83343C211yesmaximal, characteristic
3#71835C211yesmaximal, characteristic, complemented
4#70426C211yesmaximal, characteristic
5#69867C211yesmaximal, characteristic
6#67704C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9054—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#424—21no—
15C2×C3×C4⋊C8C2211yescharacteristic
16C3×#405—21no—
17C3×C4×M16C2211yescharacteristic
18C2×#825—21no—
19C2×#825—21no—
20#5088C411yescharacteristic, complemented
21#5085C411yescharacteristic, complemented
22#5080C411yescharacteristic, complemented
23#5079C411yescharacteristic, complemented
24#5048—22no—
25C4×#215—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4×C3⋊C8C2211yescharacteristic
29#4538—22no—
30#4483—21no—
31#4483—21no—
32#3167—31no—
33#2962D311yescharacteristic, complemented
34#1980—31no—
35#1503—31no—
36#1480—31no—
37#1279—31no—
38#1278—31no—
39C2×#235—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8D412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#215—22no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3—21no—
56C2×C4×Dic3—21no—
57C22×C3⋊C8—21no—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C3×C4⋊C8—22no—
64C3×C4⋊C8—22no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D411yescharacteristic
67C3×C4⋊C8D411yescharacteristic
68C3×#97—22no—
69C3×#97—21no—
70C3×C4×C8—22no—
71C3×C4×C8—21no—
72#835—22no—
73C3⋊#95—21no—
74#830—21no—
75#825—42no—
76#825—42no—
77#809—22no—
78#808—22no—
79C4⋊(C3⋊C8)—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84#533C2×D311yescharacteristic
85#532—31no—
86#498—31no—
87#492—31no—
88#487—31no—
89#486—31no—
90#465—62no—
91#446—22no—
92#446—21no—
93C8×Q8—21no—
94C4×SD16—62no—
95#431—31no—
96#424—21no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#104—61no—
99C2×#104—61no—
100#405—21no—
101C4×M16C2×D311yescharacteristic
102C2×C4×C8—31no—
103C4×C4⋊C4—31no—
104#328—62no—
105D4⋊C8—61no—
106D4⋊C8—61no—
107C2×C3×Q8—21no—
108C2×C3×Q8C22⋊C412yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113C22×Dic3—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—21no—
126C3×M16SD1612yescomplemented
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4C22⋊C412yes—
139C3×C22⋊C4—22no—
140C3×C42—22no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#215—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—42no—
157C2×C3⋊C8—41no—
158C2×#41—61no—
159C2×SD16—62no—
160C2×M16#6711yescharacteristic
161C2×M16#6711yescharacteristic
162C22×C8—21no—
163C22×C8—61no—
164C4⋊Q8—31no—
165C4⋊Q8C4×D311yescharacteristic
166C4⋊Q8C4×D311yescharacteristic
167#127—31no—
168#125—21no—
169#124—61no—
170C22⋊Q8—21no—
171C22⋊Q8D1212yes—
172#121—61no—
173#121—61no—
174C4×Q8—21no—
175C4×Q8#6712yes—
176C4×D4—61no—
177C4×D4—61no—
178#117—21no—
179C2×C4⋊C4—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#106—62no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—61no—
187C4⋊C8#6711yescharacteristic
188C4⋊C8#6711yescharacteristic
189#104—122no—
190#104—122no—
191#103—62no—
192#102—62no—
193C22⋊C8—61no—
194#97—22no—
195#97—21no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—31no—
199C4×C8—31no—
200C4×C8—61no—
201C4×C8—61no—
202#95—61no—
203C23×C3#12111yescharacteristic
204C22×D3—41no—
205C3×Q8—42no—
206C3×Q8—41no—
207C3×Q8—41no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4#13711yescharacteristic
223#67—81no—
224#67—81no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229D12—42no—
230D12—41no—
231D12—41no—
232D12—41no—
233D12—41no—
234C4×D3—81no—
235C4×D3—81no—
236Dic6—41no—
237Dic6—41no—
238C3×C8—22no—
239C3×C8—41no—
240C3×C8—41no—
241C3×C8—41no—
242C3⋊C8—22no—
243C3⋊C8—41no—
244C3⋊C8—81no—
245#41—121no—
246#41—121no—
247#41—121no—
248#41—121no—
249C2×Q8—21no—
250C2×Q8—61no—
251C2×Q8(C2×D3)⋊C412yes—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4C2×D1211yescharacteristic
259C22×C4#23511yescharacteristic
260C22×C4C2×#6711yescharacteristic
261SD16—124no—
262M16—21no—
263M16—21no—
264M16—21no—
265M16#21512yescomplemented
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—122no—
275C2×C8—121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—121no—
285C4⋊C4(C2×D3)⋊C412yes—
286C22⋊C4—22no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C42—22no—
290C42—121no—
291C42—121no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42C2×#6711yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#46011yescharacteristic
300C22×C3#46411yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4C4×SD1611yescharacteristic
310C3×C4#44011yescharacteristic
311C3×C4#44311yescharacteristic
312C3×C4#44411yescharacteristic
313C3×C4#44811yescharacteristic
314C3×C4#44911yescharacteristic
315Dic3—82no—
316Dic3—81no—
317C23—121no—
318C23#93011yescharacteristic
319Q8—42no—
320Q8—41no—
321Q8—41no—
322Q8—121no—
323Q8—121no—
324D4—122no—
325D4—121no—
326D4—121no—
327D4—121no—
328D4—121no—
329D4—241no—
330D4—241no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—122no—
337C2×C4—121no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4C2×(C2×D3)⋊C411yescharacteristic
343C2×C4C4×#6711yescharacteristic
344C2×C4C4×#6711yescharacteristic
345C2×C4#92911yescharacteristic
346C2×C4#93011yescharacteristic
347C2×C4C2×#21511yescharacteristic
348C2×C4#94211yescharacteristic
349C2×C4#94911yescharacteristic
350C2×C4#95011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—22no—
353C8—41no—
354C8—41no—
355C8—41no—
356C8—62no—
357C8—121no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3#187611yescharacteristic
362C2×C3#188011yescharacteristic
363C2×C3#188411yescharacteristic
364D3—161no—
365C22—21no—
366C22—21no—
367C22—241no—
368C22—241no—
369C22—241no—
370C22#502711yescharacteristic
371C22#502811yescharacteristic
372C22#521311yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—242no—
379C4—241no—
380C4C4×#21511yescharacteristic
381C4#502611yescharacteristic
382C4#514411yescharacteristic
383C4#514511yescharacteristic
384C4#522711yescharacteristic
385C4#523811yescharacteristic
386C3#905411yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—481no—
390C2#7163311yescharacteristic
391C2#7201011yescharacteristic
392C2#7202111yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2962C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2962D3
hypocenterC3#9054
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. C3×#2962C2semidirect
3. #71835C2semidirect
6. #67704C2semidirect
7. #67667C2semidirect
20. #5088C4semidirect
21. #5085C4semidirect
22. #5080C4semidirect
23. #5079C4semidirect
33. #2962D3semidirect
126. C3×M16SD16semidirect
265. M16#215semidirect
386. C3#9054semidirect

Maximal subgroups

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

subgroupindex
C3×#29622
#833432
#718352
#704262
#698672
#677042
#676672
#90543