Finite Group Databasebeta

← #10690971

Subgroups — #10690971

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73999C211yesmaximal, characteristic
2C2×#4651C211yesmaximal, characteristic, complemented
3C2×#4470C211yesmaximal, characteristic
4#69910C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9157—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4×Dic3C2211yescharacteristic
11C2×#886—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4988C412yescomplemented
20#4984C412yescomplemented
21#4958—21no—
22#4670—22no—
23#4651—22no—
24#4651—22no—
25#4550—22no—
26#4470—22no—
27#4451—22no—
28#2854—31no—
29C2×#383—31no—
30#1817—31no—
31C2×#327—31no—
32C2×#327—31no—
33#1294—31no—
34#1294D311yescharacteristic, complemented
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C22×#67—21no—
38C23×Dic3—21no—
39C23×Dic3D411yescharacteristic
40C23×Dic3D411yescharacteristic
41C22×D12C2×C411yescharacteristic
42C22×C3×C8—21no—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C22⋊C4—21no—
45#930—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3C2×C412yes—
59C2×C4×Dic3D412yes—
60C22×C3⋊C8—21no—
61#886—42no—
62#886—42no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95D412yes—
68C3⋊#95—24no—
69C3⋊#95—24no—
70C3⋊#95—21no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×#123—61no—
76C2×#121—31no—
77C22×C42—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8C2×D311yescharacteristic
80#390—32no—
81#389—32no—
82#388—62no—
83#386—61no—
84#383—62no—
85#383—62no—
86#380C2×D311yescharacteristic
87C2×#95—31no—
88C2×#95—61no—
89C2×#95—61no—
90C2×#95C2×D311yescharacteristic
91#336—62no—
92#327—62no—
93#327—62no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110C2×#67—41no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C22×Dic3—41no—
129C22×Dic3—41no—
130C22×Dic3C22⋊C412yes—
131C22×Dic3C22⋊C412yes—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×C3×C8—42no—
136C2×C3×C8—41no—
137C3×C4⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140(C2×D3)⋊C4—44no—
141(C2×D3)⋊C4—44no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—44no—
147C4×Dic3—22no—
148C4×Dic3—22no—
149C4×Dic3—22no—
150C4×Dic3—22no—
151C2×C3⋊C8—42no—
152C2×C3⋊C8—41no—
153C22×D4—31no—
154C22×D4—61no—
155C23×C4—21no—
156C23×C4—31no—
157C23×C4—31no—
158C23×C4—61no—
159C23×C4C22×D311yescharacteristic
160C22×C8—21no—
161C22×C8—61no—
162#123—122no—
163#123—122no—
164#121—61no—
165#121—61no—
166#121—61no—
167#121—61no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4C4×D312yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—61no—
175C2×C42—32no—
176C2×C42—32no—
177C2×C42—61no—
178C2×C42—61no—
179C2×C42—61no—
180C2×C42—61no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185#95—21no—
186#95—21no—
187#95—32no—
188#95—62no—
189#95—64no—
190#95—64no—
191#95—61no—
192#95D1212yes—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3C2×C22⋊C411yescharacteristic
196C23×C3#12011yescharacteristic
197C23×C3#12311yescharacteristic
198C23×C3#13511yescharacteristic
199C23×C3#13511yescharacteristic
200C23×C3AGL(1,ℤ/8)11yescharacteristic
201C23×C3AGL(1,ℤ/8)11yescharacteristic
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—42no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—82no—
220C2×C3×C4—82no—
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225#67—81no—
226#67—81no—
227#67—81no—
228#67—81no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—82no—
248C2×Dic3—82no—
249D12—81no—
250D12—81no—
251C3×C8—81no—
252C3⋊C8—81no—
253C24—61no—
254C24D3×D411yescharacteristic
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—122no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—62no—
283C22×C4—122no—
284C22×C4—122no—
285C22×C4—122no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4#23511yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4#23911yescharacteristic
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—42no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—124no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—124no—
305C22⋊C4—124no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—62no—
309C42—62no—
310C42—62no—
311C42—62no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3#38611yescharacteristic
321C22×C3#41012yes—
322C22×C3#41611yescharacteristic
323C22×C3#41811yescharacteristic
324C22×C3#42112yes—
325C22×C3#44911yescharacteristic
326C22×C3#44911yescharacteristic
327C22×C3#48211yescharacteristic
328C22×C3#48211yescharacteristic
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—82no—
333C2×D3—82no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—82no—
339C3×C4—82no—
340Dic3—42no—
341Dic3—42no—
342Dic3—82no—
343Dic3—81no—
344C23—22no—
345C23—22no—
346C23—121no—
347C23—122no—
348C23—121no—
349C23—121no—
350C23—122no—
351C23D3×C22⋊C411yescharacteristic
352C23#88211yescharacteristic
353C23#89411yescharacteristic
354C23#91111yescharacteristic
355C23#91411yescharacteristic
356C23#91611yescharacteristic
357C23#91911yescharacteristic
358D4—241no—
359D4—241no—
360D4—241no—
361D4—241no—
362D4—241no—
363D4—241no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—242no—
397C2×C4—242no—
398C2×C4C4×D1211yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#89311yescharacteristic
401C2×C4#89511yescharacteristic
402C8—81no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3#151412yes—
409C2×C3#152112yes—
410C2×C3#187511yescharacteristic
411C2×C3#189711yescharacteristic
412C2×C3#189911yescharacteristic
413D3—161no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—242no—
426C22—242no—
427C22#466511yescharacteristic
428C22#474012yes—
429C22#477011yescharacteristic
430C22#478111yescharacteristic
431C22#480411yescharacteristic
432C22#481011yescharacteristic
433C22#482112yes—
434C22#486611yescharacteristic
435C22#488511yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—122no—
443C4—122no—
444C4—242no—
445C4—241no—
446C3#915711yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—481no—
452C2#6828112yes—
453C2#6831512yes—
454C2#6875711yescharacteristic
455C2#6888111yescharacteristic
456C2#6895411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9157
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. C2×#4651C2semidirect
4. #69910C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
19. #4988C4semidirect
20. #4984C4semidirect
34. #1294D3semidirect
446. C3#9157semidirect

Maximal subgroups

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

subgroupindex
#739992
C2×#46512
C2×#44702
#699102
C2×#44512
C3×#12942
#676242
#91573