Finite Group Databasebeta

← #10694567

Subgroups — #10694567

Order 768.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83334C211yesmaximal, characteristic
2C3×#1518C211yesmaximal, characteristic, complemented
3#71838C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic
5#70740C211yesmaximal, characteristic, complemented
6#69547C211yesmaximal, characteristic, complemented
7#69545C211yesmaximal, characteristic
8#9757—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5093C411yescharacteristic, complemented
15#5088C411yescharacteristic, complemented
16#5088C411yescharacteristic, complemented
17#5068—21no—
18#5068—21no—
19#5002—21no—
20#5002—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#328—22no—
25#4538—22no—
26#4529—21no—
27#4529—21no—
28#4470—22no—
29#4454—21no—
30C8×C3⋊C8—21no—
31#4451—22no—
32#3131—31no—
33#1835—31no—
34#1518—31no—
35#1518D311yescharacteristic, complemented
36#1517—31no—
37#1517—31no—
38#1450—31no—
39C2×#235—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3—21no—
53C22×C3⋊C8D411yescharacteristic
54C22×C3⋊C8D411yescharacteristic
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#869C2×C411yescharacteristic
60C3×C4⋊C8—21no—
61C3×C22⋊C8D412yes—
62C3×C4×C8—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830D412yes—
68#810—21no—
69#810—21no—
70#810—21no—
71#810—21no—
72#809—21no—
73#809—21no—
74#809—21no—
75#809—21no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#533C2×D311yescharacteristic
83#532—31no—
84#488—61no—
85#488—61no—
86#487—31no—
87#487—31no—
88#486—31no—
89#486—31no—
90#431—31no—
91#431C2×D311yescharacteristic
92#416—61no—
93#416—61no—
94C2×C4×C8—31no—
95#382—31no—
96#382C2×D311yescharacteristic
97#340—61no—
98#340—61no—
99#328—22no—
100#328—62no—
101#327—62no—
102#327—62no—
103#322—61no—
104C82—61no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C2×#67—41no—
112C22×Dic3—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×C3×C8—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×C3⋊C8C22⋊C411yescharacteristic
152C2×#41—61no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—31no—
156C4⋊Q8C4×D311yescharacteristic
157C4⋊Q8C4×D311yescharacteristic
158#127—31no—
159#125—21no—
160#124—61no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163#121—61no—
164#121—61no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×D4—61no—
168C4×D4—61no—
169#117—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—61no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—61no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179#102—61no—
180#102—61no—
181#102—61no—
182#102—61no—
183C22⋊C8—32no—
184C22⋊C8#6712yes—
185C4×C8—21no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—31no—
189C4×C8—31no—
190C4×C8—61no—
191#95—21no—
192#95—21no—
193#95—61no—
194#95—61no—
195C23×C3#12711yescharacteristic
196C22×D3—41no—
197C3×Q8—81no—
198C3×Q8—81no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12411yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216#67—81no—
217#67—81no—
218C2×Dic3—41no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222D12—41no—
223D12—41no—
224D12—81no—
225D12—81no—
226C4×D3—81no—
227C4×D3—81no—
228Dic6—41no—
229Dic6—41no—
230C3×C8—42no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—42no—
236#41—121no—
237#41—121no—
238#41—121no—
239#41—121no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—121no—
247C22×C4—41no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—22no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—62no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—41no—
279C42C2×C4×D311yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3#39011yescharacteristic
286C22×C3#49511yescharacteristic
287C22×C3#49511yescharacteristic
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4—161no—
298C3×C4#41611yescharacteristic
299C3×C4#41611yescharacteristic
300C3×C4#48811yescharacteristic
301C3×C4#48811yescharacteristic
302Dic3—81no—
303Dic3—161no—
304C23—121no—
305C23#94011yescharacteristic
306Q8—81no—
307Q8—81no—
308Q8—121no—
309Q8—121no—
310D4—121no—
311D4—121no—
312D4—241no—
313D4—241no—
314D4—241no—
315D4—241no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—121no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4D3×C22⋊C411yescharacteristic
330C2×C4#67⋊C411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88511yescharacteristic
333C2×C4#91611yescharacteristic
334C2×C4#91611yescharacteristic
335C2×C4#91911yescharacteristic
336C2×C4#91911yescharacteristic
337C2×C4C4×#6711yescharacteristic
338C2×C4#93911yescharacteristic
339C8—42no—
340C8—61no—
341C8—61no—
342C8—61no—
343C8—61no—
344C8—122no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#195811yescharacteristic
348C2×C3#197611yescharacteristic
349C2×C3#197611yescharacteristic
350D3—161no—
351C22—21no—
352C22—21no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22#495711yescharacteristic
357C22#516811yescharacteristic
358C22#519111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—81no—
365C4—161no—
366C4—241no—
367C4—481no—
368C4#480011yescharacteristic
369C4#480011yescharacteristic
370C4#480411yescharacteristic
371C4#480411yescharacteristic
372C3#975711yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—481no—
376C2#7004211yescharacteristic
377C2#7022611yescharacteristic
378C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9757
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. C3×#1518C2semidirect
3. #71838C2semidirect
5. #70740C2semidirect
6. #69547C2semidirect
13. #5093C4semidirect
14. #5093C4semidirect
15. #5088C4semidirect
16. #5088C4semidirect
35. #1518D3semidirect
372. C3#9757semidirect

Maximal subgroups

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

subgroupindex
#833342
C3×#15182
#718382
#707902
#707402
#695472
#695452
#97573