Finite Group Databasebeta

← #10696031

Subgroups — #10696031

Order 768.

598 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3103C211yesmaximal, characteristic, complemented
2#83298C211yesmaximal, characteristic
3#82492C211yesmaximal, characteristic, complemented
4#71816C211yesmaximal, characteristic, complemented
5#71622C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#481—21no—
13C3×#475—21no—
14C3×#460—21no—
15C3×#448—21no—
16C3×C4×SD16—22no—
17C3×#431C2211yescharacteristic
18C2×#837—22no—
19#5231—21no—
20#5087—21no—
21#5072—21no—
22#5036—21no—
23#5033—21no—
24C4×#213—22no—
25#4992—21no—
26#4991—21no—
27#4986—21no—
28C4×(C2×D3)⋊C4—22no—
29#4932C2211yescharacteristic
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4535—22no—
33#4459—22no—
34#3103D311yescharacteristic, complemented
35#3100—31no—
36#2398—31no—
37#1963—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×SD16—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#213—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#837—44no—
79#832—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#809—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93C2×C4×D4—31no—
94#481—21no—
95#480—61no—
96#475—21no—
97#474—61no—
98#460—21no—
99#459—61no—
100#448—21no—
101#447—61no—
102C4×SD16—22no—
103C4×D8—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423—31no—
107C2×#104—62no—
108#400—61no—
109#397—61no—
110#394—61no—
111#393—61no—
112C4×C22⋊C4—62no—
113#376—31no—
114Q8⋊C8—62no—
115D4⋊C8—62no—
116C23×D3—21no—
117C3×#41—42no—
118C3×#41—42no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—42no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—42no—
141C2×C4×D3—42no—
142C3×SD16—44no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#213—44no—
159(C2×D3)⋊C4—44no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4×Dic3—22no—
166C4×Dic3—22no—
167#208—42no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×#41—21no—
171C22×Q8#6711yescharacteristic
172C22×D4—31no—
173C23×C4—61no—
174C2×SD16—22no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8#6711yescharacteristic
178#127#6711yescharacteristic
179#124—21no—
180#121—21no—
181#121—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C42—21no—
199C2×C42—32no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—22no—
203C4⋊C8—21no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#104—124no—
208#103—22no—
209#102—22no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C4×C8—32no—
215C4×C8#6712yes—
216#95—62no—
217#95—62no—
218#95—61no—
219#95—61no—
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C22×D3—22no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—42no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—42no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—82no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12711yescharacteristic
252C2×C3×C4#13512yes—
253C2×C3×C4AGL(1,ℤ/8)12yes—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259D12—22no—
260D12—22no—
261D12—41no—
262D12—41no—
263C4×D3—82no—
264C4×D3—81no—
265C4×D3—81no—
266C3×C8—42no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C24—61no—
270#41—42no—
271#41—42no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—122no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—121no—
291C22×C4—121no—
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—44no—
296D8—124no—
297M16—122no—
298C2×C8—22no—
299C2×C8—62no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—41no—
316C42—41no—
317C42—62no—
318C42—62no—
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39211yescharacteristic
329C22×C3#44912yes—
330C2×D3—44no—
331C2×D3—42no—
332C2×D3—82no—
333C2×D3—81no—
334C2×D3—81no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—42no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4#46312yes—
343C3×C4#49511yescharacteristic
344C3×C4#49611yescharacteristic
345Dic3—82no—
346Dic3—81no—
347C23—41no—
348C23—62no—
349C23—122no—
350C23—121no—
351C23—122no—
352C23—121no—
353C23#93711yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—62no—
363D4—62no—
364D4—82no—
365D4—121no—
366D4—121no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—242no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#91412yes—
386C2×C4#91612yes—
387C2×C4#93711yescharacteristic
388C2×C4#93811yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#94011yescharacteristic
392C2×C4#95311yescharacteristic
393C8—42no—
394C8—122no—
395C8—241no—
396C2×C3—22no—
397C2×C3—161no—
398C2×C3#201011yescharacteristic
399C2×C3#201011yescharacteristic
400C2×C3#201211yescharacteristic
401D3—82no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—124no—
408C22—122no—
409C22—242no—
410C22—241no—
411C22—241no—
412C22#517212yes—
413C22#522311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—242no—
422C4—241no—
423C4#516512yes—
424C4#516811yescharacteristic
425C4#517611yescharacteristic
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—161no—
429C2—242no—
430C2#7171511yescharacteristic
431C2#7205511yescharacteristic
432C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3103C2
socleC22×C3#392
prime radical#83298C2
hypercenter#3103D3
hypocenterC3#10013
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. C3×#3103C2semidirect
3. #82492C2semidirect
4. #71816C2semidirect
5. #71622C2semidirect
6. #70762C2semidirect
7. #67737C2semidirect
34. #3103D3semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
C3×#31032
#832982
#824922
#718162
#716222
#707622
#677372
#100133