Finite Group Databasebeta

← #10691914

Subgroups — #10691914

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2328C211yesmaximal, characteristic, complemented
2#83166C211yesmaximal, characteristic
3#71759C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic, complemented
5#69440C211yesmaximal, characteristic, complemented
6#67631C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9299—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×C4×D4—21no—
11#5477C2211yescharacteristic
12C3×#390—21no—
13C3×#386—22no—
14C3×#386—22no—
15C3×#382C2211yescharacteristic
16C3×#376C2211yescharacteristic
17#5217—22no—
18#5073C412yescomplemented
19#5072—22no—
20#5070C412yescomplemented
21C2×#805—21no—
22C2×#805—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4544—22no—
27#4451—22no—
28#3205—31no—
29#2328D311yescharacteristic, complemented
30#1963—31no—
31#1545—31no—
32#1517—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4D412yes—
38C23×C3×C4—22no—
39C2×#235—21no—
40C3×#127—21no—
41C3×#127—21no—
42C3×#127C2×C412yes—
43C3×#127D412yes—
44C3×C4×D4—42no—
45C3×C4×D4—42no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C42D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53C2×#213—24no—
54#930—21no—
55#930—21no—
56C2×#208—21no—
57#875—21no—
58#874C2×C411yescharacteristic, complemented
59C4×D12—21no—
60C4×D12—21no—
61#869C2×C411yescharacteristic, complemented
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95D412yes—
66#832—24no—
67#830—21no—
68#830—21no—
69#805—42no—
70#805—42no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8C2×C412yescomplemented
77#532—31no—
78C2×#127C2×D311yescharacteristic
79C2×C4×D4—21no—
80#493—32no—
81#486—32no—
82#459—62no—
83#447—62no—
84#431—31no—
85#431—31no—
86#423—31no—
87C2×#104—61no—
88C2×#104—61no—
89#390—21no—
90#386—22no—
91#386—22no—
92#382C2×D311yescharacteristic
93#376C2×D311yescharacteristic
94#331—62no—
95#327—62no—
96C24×C3—22no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—22no—
101C2×C3×D4—22no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—82no—
107C2×C3×D4—82no—
108C2×C3×D4C22⋊C414yes—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—44no—
113C22×C3×C4—82no—
114C22×C3×C4—82no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122#235—41no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—44no—
131C3×C22⋊C4—82no—
132C3×C22⋊C4—82no—
133C3×C42—21no—
134C3×C42—21no—
135C3×C42—21no—
136C3×C42D812yescomplemented
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#213—84no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146#208—41no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×#41—61no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4#6712yes—
155C23×C4—22no—
156C2×D8—64no—
157C2×M16—61no—
158C4⋊Q8—31no—
159#127—21no—
160#127—21no—
161#127—31no—
162#127C4×D312yes—
163#127D1212yes—
164#124—61no—
165#121—61no—
166#121—61no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42#6711yescharacteristic
176C2×C42#6711yescharacteristic
177C2×C42C22×D311yescharacteristic
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#104—122no—
183#104—122no—
184#102—64no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187C4×C8—32no—
188C4×C8—61no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95#6712yes—
193C23×C3—24no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3#12011yescharacteristic
201C22×D3—41no—
202C3×D4—44no—
203C3×D4—42no—
204C3×D4—42no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—82no—
208C3×D4—82no—
209C3×D4—82no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—88no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232#67—81no—
233#67—81no—
234C2×Dic3—41no—
235D12—42no—
236D12—41no—
237D12—41no—
238D12—81no—
239C4×D3—81no—
240C4×D3—81no—
241Dic6—41no—
242Dic6—41no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C3⋊C8—81no—
246C24—22no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—61no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—121no—
267C2×D4(C2×D3)⋊C414yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—44no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—121no—
275C22×C4C2×#6711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278D8—244no—
279M16—121no—
280M16—121no—
281C2×C8—62no—
282C2×C8—121no—
283C2×C8—121no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C22⋊C4—44no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42#21312yescomplemented
297C42C2×C4×D311yescharacteristic
298C42C2×D1211yescharacteristic
299C42#23511yescharacteristic
300C42#23511yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3#38611yescharacteristic
311C22×C3#44711yescharacteristic
312C22×C3#44911yescharacteristic
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—42no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4C4×D811yescharacteristic
324C3×C4#44211yescharacteristic
325C3×C4#45911yescharacteristic
326C3×C4#46311yescharacteristic
327Dic3—81no—
328C23—24no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—121no—
336C23#95311yescharacteristic
337Q8—121no—
338Q8—121no—
339D4—44no—
340D4—42no—
341D4—42no—
342D4—42no—
343D4—42no—
344D4—82no—
345D4—82no—
346D4—82no—
347D4—122no—
348D4—121no—
349D4—121no—
350D4—241no—
351D4—241no—
352D4—241no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—88no—
365C2×C4—121no—
366C2×C4—241no—
367C2×C4—241no—
368C2×C4C2×(C2×D3)⋊C411yescharacteristic
369C2×C4C4×#6711yescharacteristic
370C2×C4C4×#6711yescharacteristic
371C2×C4#92911yescharacteristic
372C2×C4#93011yescharacteristic
373C2×C4#93011yescharacteristic
374C2×C4C2×#21311yescharacteristic
375C2×C4#93211yescharacteristic
376C2×C4#94911yescharacteristic
377C2×C4#95011yescharacteristic
378C8—122no—
379C8—241no—
380C8—241no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—84no—
384C2×C3#187911yescharacteristic
385C2×C3#190411yescharacteristic
386C2×C3#191111yescharacteristic
387D3—161no—
388C22—21no—
389C22—21no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—241no—
398C22—241no—
399C22—241no—
400C22#521311yescharacteristic
401C22#521711yescharacteristic
402C22#523811yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—82no—
409C4—81no—
410C4—241no—
411C4C4×#21311yescharacteristic
412C4#501411yescharacteristic
413C4#501511yescharacteristic
414C4#501611yescharacteristic
415C3#929911yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—84no—
419C2—481no—
420C2#7161311yescharacteristic
421C2#7165111yescharacteristic
422C2#7165211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2328C2
socleC22×C3#386
prime radical#83166C2
cosocleC2×C3×C42C23
hypercenter#2328D3
hypocenterC3#9299
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. C3×#2328C2semidirect
3. #71759C2semidirect
4. #69545C2semidirect
5. #69440C2semidirect
6. #67631C2semidirect
7. #67614C2semidirect
18. #5073C4semidirect
20. #5070C4semidirect
29. #2328D3semidirect
58. #874C2×C4semidirect
61. #869C2×C4semidirect
76. C4×C3⋊C8C2×C4semidirect
136. C3×C42D8semidirect
296. C42#213semidirect
415. C3#9299semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
#831662
#717592
#695452
#694402
#676312
#676142
#92993