Finite Group Databasebeta

← #10690668

Subgroups — #10690668

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2965C211yesmaximal, characteristic, complemented
2C4×#949C211yesmaximal, characteristic
3#71639C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic, complemented
6#67744C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9097—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#446—21no—
13C3×#436—21no—
14C3×#435—22no—
15C3×#431—21no—
16C3×#431C2211yescharacteristic
17C3×C4×M16—21no—
18C2×C3×C4×C8C2211yescharacteristic
19#5232C411yescharacteristic
20#5026—21no—
21C4×#215—21no—
22#5014—21no—
23C4×#213—21no—
24#5001C411yescharacteristic
25#4995C411yescharacteristic
26#4925C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31#4468—22no—
32#4459—21no—
33#4459—21no—
34#2965D311yescharacteristic, complemented
35C4×AGL(1,ℤ/8)—31no—
36#1477—31no—
37C2×D4⋊C8—31no—
38#1460—31no—
39#1283—31no—
40#1278—31no—
41C2×C3×#41C2×C411yescharacteristic
42C22×D12C2×C411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—22no—
45C22×C3×C8—21no—
46C3×C4×Q8D411yescharacteristic
47C3×C4×Q8D411yescharacteristic
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4D411yescharacteristic
51C3×C4×D4D411yescharacteristic
52C3×#117—21no—
53C3×#117C2×C411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56#949—22no—
57#949—22no—
58C2×#215—21no—
59C2×#213—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×#208C2×C411yescharacteristic
63C2×#208C2×C411yescharacteristic
64C22×C3⋊C8—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C3×C4⋊C8—22no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—22no—
72C3×C22⋊C8D412yes—
73C3×#97—22no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3×C4×C8D411yescharacteristic
77C3×C4×C8D411yescharacteristic
78#835—21no—
79#832—21no—
80#830—21no—
81#809—21no—
82#809—21no—
83#808—21no—
84#807—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×AGL(1,ℤ/8)—31no—
92C4×#41C2×D311yescharacteristic
93C2×C4×D4—31no—
94#446—21no—
95#442—61no—
96#440—61no—
97C4×SD16—61no—
98C4×D8—61no—
99#436—21no—
100#435—22no—
101#431—21no—
102#431—31no—
103#431C2×D311yescharacteristic
104#428—31no—
105C2×C4⋊C8—31no—
106#418—31no—
107#417—31no—
108C4×M16—21no—
109C4×M16—31no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—61no—
112D4⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115#323—62no—
116C23×D3—21no—
117C3×#41—21no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41—21no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4C22⋊C411yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12M1612yescomplemented
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—81no—
140C2×C4×D3—81no—
141C3×M16—44no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8—42no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#215—24no—
161#213—24no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166#208—21no—
167#208—21no—
168#208—21no—
169#208M1612yescomplemented
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—81no—
175C2×#41C4×D311yescharacteristic
176C22×D4—31no—
177C23×C4—61no—
178AGL(1,ℤ/8)—62no—
179AGL(1,ℤ/8)—62no—
180C2×SD16—61no—
181C2×D8—61no—
182C2×M16—22no—
183C2×M16—31no—
184C2×M16—31no—
185C22×C8—21no—
186C22×C8—61no—
187C4×Q8D1211yescharacteristic
188C4×Q8#6711yescharacteristic
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—61no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4D1211yescharacteristic
196C4×D4#6711yescharacteristic
197#117—21no—
198#117C4×D311yescharacteristic
199C2×C4⋊C4—31no—
200C2×C22⋊C4—61no—
201C2×C42—21no—
202C2×C42C22×D311yescharacteristic
203#107—61no—
204#106—61no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210#103—61no—
211#102—61no—
212#102—61no—
213#102—61no—
214C22⋊C8—22no—
215C22⋊C8—61no—
216C22⋊C8#6712yes—
217#97—22no—
218#97—61no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—61no—
222C4×C8—61no—
223C4×C8#6711yescharacteristic
224C4×C8#6711yescharacteristic
225C23×C3—21no—
226C23×C3C4×D411yescharacteristic
227C22×D3—22no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—41no—
237C3×D4—41no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12311yescharacteristic
255C2×C3×C4C2×M1611yescharacteristic
256C2×C3×C4C2×M1611yescharacteristic
257C2×C3×C4#13111yescharacteristic
258C2×C3×C4#13111yescharacteristic
259C2×Dic3—41no—
260C2×Dic3—81no—
261D12—22no—
262D12—22no—
263D12—41no—
264D12—41no—
265C4×D3—82no—
266C4×D3—81no—
267C4×D3—81no—
268C3×C8—22no—
269C3×C8—41no—
270C3×C8—42no—
271C3⋊C8—22no—
272C3⋊C8—41no—
273C3⋊C8—81no—
274C3⋊C8—81no—
275C24—61no—
276#41—21no—
277#41—21no—
278#41—21no—
279#41—21no—
280C2×Q8(C2×D3)⋊C411yescharacteristic
281C2×D4—21no—
282C2×D4—32no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4(C2×D3)⋊C411yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—61no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—241no—
296C22×C4—241no—
297C22×C4C2×C4×D311yescharacteristic
298C22×C4#23511yescharacteristic
299C22×C4C2×#6711yescharacteristic
300SD16—64no—
301D8—64no—
302M16—32no—
303M16—44no—
304M16—61no—
305M16—61no—
306M16—61no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—42no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—241no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4(C2×D3)⋊C411yescharacteristic
321C4⋊C4(C2×D3)⋊C411yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42C2×D1211yescharacteristic
331C42#23511yescharacteristic
332C42C2×#6711yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3#38611yescharacteristic
340C22×C3#43511yescharacteristic
341C22×C3#43611yescharacteristic
342C2×D3—42no—
343C2×D3—82no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—82no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4#40711yescharacteristic
354C3×C4#40811yescharacteristic
355C3×C4#43511yescharacteristic
356C3×C4#43611yescharacteristic
357C3×C4#47111yescharacteristic
358C3×C4#47211yescharacteristic
359Dic3—81no—
360C23—21no—
361C23—62no—
362C23—121no—
363C23—121no—
364C23—241no—
365C23—241no—
366C23C4×#6711yescharacteristic
367Q8—21no—
368Q8—21no—
369D4—21no—
370D4—21no—
371D4—41no—
372D4—41no—
373D4—62no—
374D4—62no—
375D4—121no—
376D4—121no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—121no—
389C2×C4—242no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4D3×M1611yescharacteristic
394C2×C4D3×M1611yescharacteristic
395C2×C4#90711yescharacteristic
396C2×C4#90711yescharacteristic
397C2×C4C2×(C2×D3)⋊C411yescharacteristic
398C2×C4C4×#6711yescharacteristic
399C2×C4#92911yescharacteristic
400C2×C4#93011yescharacteristic
401C2×C4#93011yescharacteristic
402C2×C4#95311yescharacteristic
403C8—22no—
404C8—41no—
405C8—42no—
406C8—62no—
407C8—121no—
408C8—241no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—81no—
413C2×C3#187011yescharacteristic
414C2×C3#190611yescharacteristic
415C2×C3#190711yescharacteristic
416D3—82no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—122no—
423C22—242no—
424C22—242no—
425C22—241no—
426C22—241no—
427C22#512711yescharacteristic
428C22#512811yescharacteristic
429C22#521311yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—241no—
437C4#512611yescharacteristic
438C4#512711yescharacteristic
439C4#512811yescharacteristic
440C4#512911yescharacteristic
441C4#514111yescharacteristic
442C4#514211yescharacteristic
443C3#909711yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—81no—
447C2—242no—
448C2#7164511yescharacteristic
449C2#7199611yescharacteristic
450C2#7201811yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2965C2
socleC22×C3#386
prime radicalC2×#949C4
cosocleC2×C3×C42C23
hypercenter#2965D3
hypocenterC3#9097
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. C3×#2965C2semidirect
3. #71639C2semidirect
4. #70783C2semidirect
5. C2×#4459C2semidirect
6. #67744C2semidirect
7. #67656C2semidirect
34. #2965D3semidirect
136. C2×D12M16semidirect
169. #208M16semidirect
443. C3#9097semidirect

Maximal subgroups

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

subgroupindex
C3×#29652
C4×#9492
#716392
#707832
C2×#44592
#677442
#676562
#90973