Finite Group Databasebeta

← #10696048

Subgroups — #10696048

Order 768.

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

normal subgroups only

This group has 560 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×#3104C211yesmaximal, characteristic, complemented
2#83351C211yesmaximal, characteristic
3#82492C211yesmaximal, characteristic
4#71840C211yesmaximal, characteristic, complemented
5#71622C211yesmaximal, characteristic, complemented
6#70762C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#484—21no—
13C3×#477—21no—
14C3×#462—21no—
15C3×#451—21no—
16C3×C4×Q16—22no—
17C3×#431C2211yescharacteristic
18#5231—21no—
19#5084—21no—
20#5083—21no—
21#5046—21no—
22#5043—21no—
23C4×#215—22no—
24#4992—21no—
25#4991—21no—
26#4986—21no—
27C4×(C2×D3)⋊C4—22no—
28#4932C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32#4532—22no—
33#4459—22no—
34#3104D311yescharacteristic, complemented
35#3101—31no—
36#2398—31no—
37#1995—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×Q16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#215—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×#107—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#103—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#835—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#809—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4—31no—
93#484—21no—
94#483—61no—
95#477—21no—
96#476—61no—
97#462—21no—
98#461—61no—
99#451—21no—
100#450—61no—
101C4×Q16—22no—
102C4×SD16—62no—
103#431—31no—
104#431C2×D311yescharacteristic
105#423—31no—
106#400—61no—
107#397—61no—
108#394—61no—
109#393—61no—
110C4×C22⋊C4—62no—
111#376—31no—
112#339—62no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—42no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—42no—
134C2×C4×D3—42no—
135C3×Q16—44no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#215—44no—
156(C2×D3)⋊C4—44no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4×Dic3—22no—
163C4×Dic3—22no—
164#208—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8#6711yescharacteristic
168C22×D4—31no—
169C23×C4—61no—
170C2×Q16—22no—
171C2×SD16—62no—
172C2×M16—61no—
173C4⋊Q8#6711yescharacteristic
174C4⋊Q8#6711yescharacteristic
175#125—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—32no—
197C2×C42C22×D311yescharacteristic
198#107—22no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—22no—
205#103—22no—
206#103—62no—
207#102—62no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C4×C8—32no—
211C4×C8#6712yes—
212#95—62no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3#12011yescharacteristic
217C22×D3—22no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C3×Q8—22no—
223C3×Q8—22no—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—42no—
227C3×Q8—42no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4#13512yes—
244C2×C3×C4#13712yes—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250D12—22no—
251D12—22no—
252D12—41no—
253D12—41no—
254C4×D3—82no—
255C4×D3—81no—
256C4×D3—81no—
257C3×C8—42no—
258C3⋊C8—42no—
259C3⋊C8—81no—
260C24—61no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4C2×#6711yescharacteristic
280Q16—44no—
281SD16—124no—
282M16—122no—
283C2×C8—22no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—41no—
299C22⋊C4—124no—
300C22⋊C4—124no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—62no—
307C42—62no—
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C42C2×#6711yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#39211yescharacteristic
315C22×C3#45212yes—
316C2×D3—44no—
317C2×D3—42no—
318C2×D3—82no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—42no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4#46412yes—
330C3×C4#49511yescharacteristic
331C3×C4#49711yescharacteristic
332Dic3—82no—
333Dic3—81no—
334C23—62no—
335C23—122no—
336C23—121no—
337C23—122no—
338C23—121no—
339C23#93711yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—42no—
345Q8—42no—
346D4—62no—
347D4—62no—
348D4—121no—
349D4—121no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—242no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#91812yes—
368C2×C4#91912yes—
369C2×C4#93711yescharacteristic
370C2×C4#93811yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#93911yescharacteristic
373C2×C4#94011yescharacteristic
374C2×C4#95311yescharacteristic
375C8—42no—
376C8—122no—
377C8—241no—
378C2×C3—22no—
379C2×C3#201011yescharacteristic
380C2×C3#201011yescharacteristic
381C2×C3#201511yescharacteristic
382D3—82no—
383C22—21no—
384C22—21no—
385C22—124no—
386C22—122no—
387C22—242no—
388C22—241no—
389C22—241no—
390C22#518712yes—
391C22#522311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—242no—
401C4—241no—
402C4#518512yes—
403C4#519011yescharacteristic
404C4#519111yescharacteristic
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—242no—
408C2#7171511yescharacteristic
409C2#7207211yescharacteristic
410C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3104C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3104D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
1. C3×#3104C2semidirect
4. #71840C2semidirect
5. #71622C2semidirect
7. #67737C2semidirect
34. #3104D3semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
C3×#31042
#833512
#824922
#718402
#716222
#707622
#677372
#100163