Finite Group Databasebeta

← #10696043

Subgroups — #10696043

Order 768.

696 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3101C211yesmaximal, characteristic, complemented
2#83351C211yesmaximal, characteristic, complemented
3#82347C211yesmaximal, characteristic
4#71840C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10015—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×C4×#67—22no—
12C2×C4×D12C2211yescharacteristic
13C3×#483—21no—
14C3×#476—21no—
15C3×#461—21no—
16C3×#450—21no—
17C3×C4×SD16—22no—
18C3×#431C2211yescharacteristic
19#5223—21no—
20#5084—21no—
21#5083—21no—
22#5046—21no—
23#5043—21no—
24C4×#215—22no—
25#4993—21no—
26#4960—21no—
27#4949—21no—
28#4932C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4532—22no—
33#4459—22no—
34#3101—31no—
35#3101D311yescharacteristic, complemented
36#2397—31no—
37#1995—31no—
38#1486—31no—
39#1486—31no—
40#1274—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C2×C3×SD16—22no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#215—22no—
64C4×#67—44no—
65C4×#67—44no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×Dic3⋊C4—22no—
70C2×C4×Dic3D412yes—
71C2×#208—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C3×#106—22no—
77C3×C4⋊C8—21no—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C4×C8D412yescomplemented
82#835—22no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85#830—21no—
86#809—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8D412yescomplemented
92#533C2×D311yescharacteristic
93C2×C4×D4—31no—
94C2×C4×D4—62no—
95C2×C4×D4C2×D311yescharacteristic
96#483—21no—
97#483—61no—
98#476—21no—
99#476—61no—
100#461—21no—
101#461—61no—
102#450—21no—
103#450—61no—
104C4×SD16—22no—
105C4×SD16—62no—
106#431—31no—
107#431C2×D311yescharacteristic
108#423—31no—
109#399—61no—
110#399—61no—
111#392—61no—
112#392—61no—
113#376—31no—
114#339—62no—
115D4⋊C8—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C23×D3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—44no—
133C2×#67—44no—
134C2×#67—44no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—42no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×C4×D3—42no—
143C2×C4×D3—42no—
144C3×SD16—44no—
145C2×C3×C8—22no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C42—41no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162#217—44no—
163#215—44no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—44no—
170C4×Dic3—22no—
171C4×Dic3SD1614yescomplemented
172#208—42no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C22×D4—31no—
176C22×D4—62no—
177C22×D4#6711yescharacteristic
178C23×C4—21no—
179C23×C4—61no—
180C2×SD16—22no—
181C2×SD16—62no—
182C2×M16—61no—
183C4⋊Q8#6711yescharacteristic, complemented
184C4⋊Q8#6711yescharacteristic, complemented
185#125—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197C4×D4—61no—
198C4×D4—124no—
199C4×D4—124no—
200#117—21no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—61no—
208C2×C42—32no—
209C2×C42C22×D311yescharacteristic
210#106—22no—
211#106—62no—
212C4⋊C8—21no—
213C4⋊C8—61no—
214C4⋊C8—61no—
215C4⋊C8—61no—
216#103—22no—
217#103—62no—
218#102—22no—
219#102—62no—
220C22⋊C8—21no—
221C22⋊C8—61no—
222C4×C8—32no—
223C4×C8#6712yescomplemented
224#95—61no—
225#95—61no—
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3#12011yescharacteristic
232C22×D3—22no—
233C22×D3—41no—
234C22×D3—42no—
235C22×D3—41no—
236C22×D3—42no—
237C3×Q8—42no—
238C3×Q8—42no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—41no—
242C3×D4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×C3×C4C2×SD1612yes—
259C2×C3×C4AGL(1,ℤ/8)12yes—
260#67—416no—
261C2×Dic3—24no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266D12—22no—
267D12—22no—
268D12—41no—
269D12—41no—
270C4×D3—82no—
271C4×D3—81no—
272C4×D3—81no—
273C3×C8—42no—
274C3⋊C8—42no—
275C3⋊C8—81no—
276C24—21no—
277C24—61no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—122no—
286C2×D4—124no—
287C2×D4—124no—
288C2×D4—124no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—62no—
295C22×C4—61no—
296C22×C4—122no—
297C22×C4—122no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4C2×#6711yescharacteristic
303SD16—44no—
304SD16—124no—
305M16—122no—
306C2×C8—22no—
307C2×C8—62no—
308C2×C8—121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—124no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—124no—
324C22⋊C4—124no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—34no—
328C42—41no—
329C42—62no—
330C42D3×D411yescharacteristic
331C42#23711yescharacteristic
332C42C2×#6711yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—82no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3#39211yescharacteristic
342C22×C3#45012yes—
343C2×D3—44no—
344C2×D3—42no—
345C2×D3—82no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4—81no—
355C3×C4#46012yes—
356C3×C4#49211yescharacteristic
357C3×C4#49611yescharacteristic
358Dic3—44no—
359Dic3—81no—
360C23—22no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—62no—
366C23—122no—
367C23—121no—
368C23—122no—
369C23—121no—
370C23#93711yescharacteristic
371Q8—42no—
372Q8—42no—
373D4—22no—
374D4—22no—
375D4—41no—
376D4—41no—
377D4—62no—
378D4—62no—
379D4—121no—
380D4—121no—
381D4—1216no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—64no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—242no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4D3×SD1612yes—
400C2×C4#91412yes—
401C2×C4#93711yescharacteristic
402C2×C4#93811yescharacteristic
403C2×C4#93911yescharacteristic
404C2×C4#93911yescharacteristic
405C2×C4#94011yescharacteristic
406C2×C4#95311yescharacteristic
407C8—42no—
408C8—122no—
409C8—241no—
410C2×C3—22no—
411C2×C3—82no—
412C2×C3#201011yescharacteristic
413C2×C3#201011yescharacteristic
414C2×C3#201411yescharacteristic
415D3—82no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—42no—
420C22—82no—
421C22—81no—
422C22—81no—
423C22—124no—
424C22—122no—
425C22—242no—
426C22—241no—
427C22—241no—
428C22#516912yes—
429C22#522311yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4—124no—
438C4—241no—
439C4#516312yes—
440C4#517511yescharacteristic
441C4#517611yescharacteristic
442C3#1001511yescharacteristic, complemented, Sylow
443C2—22no—
444C2—82no—
445C2—242no—
446C2#7171511yescharacteristic
447C2#7205611yescharacteristic
448C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#392
prime radical#82347C2
hypercenter#3101D3
hypocenterC3#10015
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. C3×#3101C2semidirect
2. #83351C2semidirect
4. #71840C2semidirect
5. #71598C2semidirect
6. #70762C2semidirect
7. #67737C2semidirect
35. #3101D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
81. C3×C4×C8D4semidirect
91. C4×C3⋊C8D4semidirect
171. C4×Dic3SD16semidirect
183. C4⋊Q8#67semidirect
184. C4⋊Q8#67semidirect
223. C4×C8#67semidirect
442. C3#10015semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
#833512
#823472
#718402
#715982
#707622
#677372
#100153