Finite Group Databasebeta

← #10696568

Subgroups — #10696568

Order 768.

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

normal subgroups only

This group has 590 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#83264C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#71776C211yesmaximal, characteristic, complemented
6#69860C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10094—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—21no—
12C2×#944—21no—
13C2×#875C2211yescharacteristic
14C3×#489—22no—
15C3×#487—21no—
16C3×#486—21no—
17C3×#417—22no—
18C3×C4×M16C2211yescharacteristic
19#5089—21no—
20#5074—21no—
21#5040—21no—
22#5036—21no—
23#5016—22no—
24#4993—22no—
25#4993—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4547—22no—
30#4537—22no—
31#4464—22no—
32#3213—31no—
33#3141D311yescharacteristic, complemented
34#2658—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
58#947—44no—
59#944—44no—
60C2×#214—22no—
61C2×#213—22no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×Dic3⋊C4—24no—
66C2×C4×Dic3—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70#875—21no—
71C3×#103—24no—
72C3×#102—24no—
73C3×#97D412yes—
74C3×C4×C8—21no—
75C3⋊#95—21no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79#810—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83#803—21no—
84#803—21no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89C2×#124—61no—
90#496—61no—
91#489—22no—
92#489—61no—
93#487—21no—
94#486—21no—
95#463—62no—
96#448—61no—
97#447—61no—
98#432—31no—
99#432—31no—
100#417—22no—
101C4×M16C2×D311yescharacteristic
102#399—62no—
103#399—62no—
104#376—31no—
105#332—62no—
106#329—62no—
107#327—62no—
108C23×D3—21no—
109C3×#41—44no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—42no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—42no—
132C3×M16—42no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#214—82no—
147#213—82no—
148(C2×D3)⋊C4—48no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153Dic3⋊C4—48no—
154C4×Dic3—24no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×#41—21no—
158C22×Q8—21no—
159C22×Q8—31no—
160C22×D4—31no—
161C2×SD16—62no—
162C2×D8—62no—
163C2×M16—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—124no—
169C4⋊Q8#6711yescharacteristic
170#127#6711yescharacteristic
171#124—21no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176#124—124no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4#6712yes—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—32no—
190C4⋊C8—61no—
191#103—24no—
192#103—62no—
193#102—24no—
194#102—62no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#97—61no—
198#97—61no—
199#97#6712yes—
200C4×C8—21no—
201#95—32no—
202#95—61no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—41no—
208C22×D3—42no—
209C3×Q8—42no—
210C3×Q8—82no—
211C3×Q8—82no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—81no—
215C3×D4—82no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×Dic3—24no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235D12—42no—
236D12—81no—
237Dic6—42no—
238Dic6—81no—
239C3×C8—42no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—61no—
243#41—44no—
244C2×Q8—21no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—122no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—62no—
262C22×C4—61no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—242no—
267D8—242no—
268M16—42no—
269C2×C8—22no—
270C2×C8—62no—
271C2×C8—121no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—128no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—128no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C42—64no—
287C42D3×D411yescharacteristic
288C42#23711yescharacteristic
289C42C2×#6711yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3#39211yescharacteristic
297C22×C3#45412yes—
298C2×D3—84no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—81no—
309C3×C4#45312yes—
310Dic3—44no—
311Dic3—81no—
312C23—41no—
313C23—122no—
314C23—122no—
315C23—121no—
316C23—122no—
317C23#93711yescharacteristic
318Q8—42no—
319Q8—82no—
320Q8—82no—
321Q8—122no—
322Q8—241no—
323D4—42no—
324D4—42no—
325D4—81no—
326D4—82no—
327D4—122no—
328D4—241no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—64no—
337C2×C4—82no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4#93711yescharacteristic
343C2×C4#93811yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#94011yescharacteristic
347C2×C4#95311yescharacteristic
348C8—42no—
349C8—122no—
350C8—241no—
351C2×C3—22no—
352C2×C3—161no—
353C2×C3#200211yescharacteristic
354C2×C3#200311yescharacteristic
355C2×C3#200911yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—244no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22#506212yes—
367C22#522311yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—81no—
374C4—82no—
375C4—124no—
376C4—241no—
377C4#519812yes—
378C3#1009411yescharacteristic, complemented, Sylow
379C2—22no—
380C2—161no—
381C2—481no—
382C2#7173011yescharacteristic
383C2#7209411yescharacteristic
384C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3141C2
socleC22×C3#392
prime radical#83306C2
hypercenter#3141D3
hypocenterC3#10094
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. #83264C2semidirect
4. #71812C2semidirect
5. #71776C2semidirect
6. #69860C2semidirect
7. #67665C2semidirect
33. #3141D3semidirect
378. C3#10094semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#833062
#832642
#718122
#717762
#698602
#676652
#100943