Finite Group Databasebeta

← #10694572

Subgroups — #10694572

Order 768.

408 conjugacy classes of subgroups.

The subgroup lattice has 384 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#70742C211yesmaximal, characteristic, complemented
6#69549C211yesmaximal, characteristic, complemented
7#69545C211yesmaximal, characteristic
8#9758—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—
21C2×#808—21no—
22C2×#807—21no—
23#4936C2211yescharacteristic
24#4925C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#328—22no—
27#4538—22no—
28#4470—22no—
29#4455—21no—
30#4454—21no—
31#4451—22no—
32#3131—31no—
33#1846—31no—
34#1518—31no—
35#1518D311yescharacteristic, complemented
36#1517—31no—
37#1517—31no—
38#1452—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—
76#808—42no—
77#807—42no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8C2×C411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83C4×C3⋊C8D411yescharacteristic
84#533C2×D311yescharacteristic
85#532—31no—
86#488—61no—
87#488—61no—
88#487—31no—
89#487—31no—
90#486—31no—
91#486—31no—
92#431—31no—
93#431C2×D311yescharacteristic
94C2×#107—61no—
95C2×#106—61no—
96#416—61no—
97#416—61no—
98C2×C4×C8—31no—
99#382—31no—
100#382C2×D311yescharacteristic
101#328—22no—
102#328—62no—
103#327—62no—
104#327—62no—
105#322—61no—
106#322—61no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118#235—41no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—41no—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×C3⋊C8C22⋊C411yescharacteristic
152C2×C3⋊C8C22⋊C411yescharacteristic
153C2×C3⋊C8C22⋊C411yescharacteristic
154C2×#41—61no—
155C22×C8—31no—
156C22×C8—31no—
157C4⋊Q8—31no—
158C4⋊Q8C4×D311yescharacteristic
159C4⋊Q8C4×D311yescharacteristic
160#127—31no—
161#125—21no—
162#124—61no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165#121—61no—
166#121—61no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—61no—
170C4×D4—61no—
171#117—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—61no—
174C2×C42C22×D311yescharacteristic
175#107—122no—
176#106—122no—
177C4⋊C8—21no—
178C4⋊C8—61no—
179#103—61no—
180#103—61no—
181#103—61no—
182#103—61no—
183#102—61no—
184#102—61no—
185#102—61no—
186#102—61no—
187C22⋊C8—32no—
188C22⋊C8#6712yes—
189C4×C8—21no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—31no—
193C4×C8—31no—
194C4×C8—61no—
195#95—21no—
196#95—21no—
197#95—61no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C22×D3—41no—
201C3×Q8—81no—
202C3×Q8—81no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12411yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×C3×C4#13711yescharacteristic
220#67—81no—
221#67—81no—
222C2×Dic3—41no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226D12—41no—
227D12—41no—
228D12—81no—
229D12—81no—
230C4×D3—81no—
231C4×D3—81no—
232Dic6—41no—
233Dic6—41no—
234C3×C8—42no—
235C3⋊C8—21no—
236C3⋊C8—21no—
237C3⋊C8—21no—
238C3⋊C8—21no—
239C3⋊C8—42no—
240#41—121no—
241#41—121no—
242#41—121no—
243#41—121no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—41no—
252C22×C4—121no—
253C22×C4—121no—
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C2×C8—22no—
258C2×C8—31no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8—31no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—62no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—41no—
283C42C2×C4×D311yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42#23711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39011yescharacteristic
290C22×C3#49611yescharacteristic
291C22×C3#49711yescharacteristic
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4—161no—
302C3×C4#41711yescharacteristic
303C3×C4#41711yescharacteristic
304C3×C4#48911yescharacteristic
305C3×C4#48911yescharacteristic
306Dic3—81no—
307Dic3—161no—
308C23—121no—
309C23#94011yescharacteristic
310Q8—81no—
311Q8—81no—
312Q8—121no—
313Q8—121no—
314D4—121no—
315D4—121no—
316D4—241no—
317D4—241no—
318D4—241no—
319D4—241no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4D3×C22⋊C411yescharacteristic
334C2×C4#67⋊C411yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#88511yescharacteristic
337C2×C4#91411yescharacteristic
338C2×C4#91411yescharacteristic
339C2×C4#91811yescharacteristic
340C2×C4#91811yescharacteristic
341C2×C4C4×#6711yescharacteristic
342C2×C4#93911yescharacteristic
343C8—42no—
344C8—61no—
345C8—61no—
346C8—61no—
347C8—61no—
348C8—122no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3#195911yescharacteristic
352C2×C3#197611yescharacteristic
353C2×C3#197611yescharacteristic
354D3—161no—
355C22—21no—
356C22—21no—
357C22—241no—
358C22—241no—
359C22—241no—
360C22#495711yescharacteristic
361C22#517611yescharacteristic
362C22#519011yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—161no—
370C4—241no—
371C4—481no—
372C4#479411yescharacteristic
373C4#479411yescharacteristic
374C4#480311yescharacteristic
375C4#480311yescharacteristic
376C3#975811yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—481no—
380C2#7004211yescharacteristic
381C2#7022711yescharacteristic
382C2#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#9758
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. #70742C2semidirect
6. #69549C2semidirect
13. #5093C4semidirect
14. #5093C4semidirect
15. #5088C4semidirect
16. #5088C4semidirect
35. #1518D3semidirect
376. C3#9758semidirect

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
#707422
#695492
#695452
#97583