Finite Group Databasebeta

← #10696554

Subgroups — #10696554

Order 768.

534 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3141C211yesmaximal, characteristic, complemented
2#83306C211yesmaximal, characteristic
3#83263C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71776C211yesmaximal, characteristic, complemented
6#69860C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10092—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#489—22no—
13C3×#487—21no—
14C3×#486—21no—
15C3×#417—22no—
16C3×C4×M16C2211yescharacteristic
17#5231—22no—
18#5089—21no—
19#5074—21no—
20#5040—21no—
21#5036—21no—
22#5016—22no—
23#4992—22no—
24#4988—21no—
25#4973—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4537—22no—
31#4464—22no—
32#3213—31no—
33#3141D311yescharacteristic, complemented
34#2640—31no—
35#1556—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C42C2311yescharacteristic
58C2×#214—22no—
59C2×#213—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—24no—
63C2×C4×Dic3—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#875—21no—
68C3×#103—24no—
69C3×#102—24no—
70C3×#97D412yes—
71C3×C4×C8—21no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#810—22no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#496—61no—
87#489—22no—
88#489—61no—
89#487—21no—
90#486—21no—
91#463—62no—
92#448—61no—
93#447—61no—
94#432—31no—
95#432—31no—
96#417—22no—
97C4×M16C2×D311yescharacteristic
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#332—62no—
104#329—62no—
105#327—62no—
106C23×D3—21no—
107C3×#41—44no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—22no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—22no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—42no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—42no—
129C3×M16—42no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#214—82no—
144#213—82no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—84no—
150C4×Dic3—42no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×#41—21no—
154C22×Q8—21no—
155C22×Q8—31no—
156C22×D4—31no—
157C2×SD16—62no—
158C2×D8—62no—
159C2×M16—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8#6711yescharacteristic
165#127#6711yescharacteristic
166#124—21no—
167#124—61no—
168#124—61no—
169#124—61no—
170#124—61no—
171#121—21no—
172#121—21no—
173C4×D4—21no—
174C4×D4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6712yes—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—64no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182C4⋊C8—32no—
183C4⋊C8—61no—
184#103—24no—
185#103—62no—
186#102—24no—
187#102—62no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—61no—
191#97—61no—
192#97#6712yes—
193C4×C8—21no—
194#95—32no—
195#95—64no—
196#95—61no—
197C23×C3—41no—
198C23×C3#12011yescharacteristic
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—41no—
202C22×D3—42no—
203C3×Q8—42no—
204C3×Q8—82no—
205C3×Q8—82no—
206C3×D4—42no—
207C3×D4—42no—
208C3×D4—81no—
209C3×D4—82no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—44no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—82no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228D12—42no—
229D12—81no—
230Dic6—42no—
231Dic6—81no—
232C3×C8—42no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C24—61no—
236#41—44no—
237C2×Q8—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—82no—
243C2×Q8—122no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—122no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4C2×#6711yescharacteristic
258SD16—242no—
259D8—242no—
260M16—42no—
261C2×C8—22no—
262C2×C8—62no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—244no—
277C42—122no—
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C42C2×#6711yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3#39211yescharacteristic
288C22×C3#45412yes—
289C2×D3—84no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4—82no—
300C3×C4#45512yes—
301Dic3—82no—
302Dic3—81no—
303C23—41no—
304C23—122no—
305C23—122no—
306C23—121no—
307C23—122no—
308C23#93711yescharacteristic
309Q8—42no—
310Q8—82no—
311Q8—82no—
312Q8—122no—
313Q8—241no—
314D4—42no—
315D4—42no—
316D4—81no—
317D4—82no—
318D4—122no—
319D4—241no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4#93711yescharacteristic
333C2×C4#93811yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#93911yescharacteristic
336C2×C4#94011yescharacteristic
337C2×C4#95311yescharacteristic
338C8—42no—
339C8—122no—
340C8—241no—
341C2×C3—22no—
342C2×C3—161no—
343C2×C3#200311yescharacteristic
344C2×C3#200411yescharacteristic
345C2×C3#200811yescharacteristic
346D3—161no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—244no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22#506212yes—
357C22#522311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—82no—
365C4—242no—
366C4—241no—
367C4#520212yes—
368C3#1009211yescharacteristic, complemented, Sylow
369C2—22no—
370C2—161no—
371C2—481no—
372C2#7173011yescharacteristic
373C2#7209911yescharacteristic
374C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3141C2
socleC22×C3#392
prime radical#83306C2
hypercenter#3141D3
hypocenterC3#10092
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. C3×#3141C2semidirect
3. #83263C2semidirect
4. #71812C2semidirect
5. #71776C2semidirect
6. #69860C2semidirect
7. #67665C2semidirect
33. #3141D3semidirect
368. C3#10092semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#833062
#832632
#718122
#717762
#698602
#676652
#100923