Finite Group Databasebeta

← #10697299

Subgroups — #10697299

Order 768.

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

normal subgroups only

This group has 508 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×#3246C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic, complemented
3#83283C211yesmaximal, characteristic
4#71841C211yesmaximal, characteristic, complemented
5#71811C211yesmaximal, characteristic
6#70459C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#476—21no—
13C3×#475—21no—
14C3×#465—22no—
15C3×#461—21no—
16C3×#460—21no—
17C2×C3×C4⋊C8C2211yescharacteristic
18#5232—22no—
19#5093—21no—
20#5092—21no—
21#5079—21no—
22#5070—21no—
23#5055—22no—
24#5050—22no—
25#5031—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4551—22no—
30#4537—22no—
31#4532—22no—
32#3246D311yescharacteristic
33#3209—31no—
34#3141—31no—
35#1987—31no—
36#1986—31no—
37#1502—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C2×C3×SD16—22no—
42C22×C3×C8D412yes—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#125—21no—
48C3×#124—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×#117—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208D412yes—
65#869—21no—
66#869—21no—
67#869—21no—
68#869D412yes—
69C3×#106—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×#103—22no—
73C3×#102—22no—
74#835—24no—
75#832—24no—
76C3⋊#95—21no—
77#830D412yes—
78#810—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#494—61no—
87#489—62no—
88#487—61no—
89#487—61no—
90#486—61no—
91#476—21no—
92#475—21no—
93#465—22no—
94#462—62no—
95#461—21no—
96#460—21no—
97#451—62no—
98#431—31no—
99C2×C4⋊C8C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#327—62no—
106C3×#41—42no—
107C3×#41—42no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—41no—
120C22×Dic3—41no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—82no—
124C3×SD16—82no—
125C2×C3×C8—22no—
126C2×C3×C8—42no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#216—84no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4×Dic3—42no—
151#208—22no—
152#208—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×#41—21no—
156C22×Q8—61no—
157C2×Q16—64no—
158C2×SD16—22no—
159C2×M16—32no—
160C22×C8#6712yes—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8#6711yescharacteristic
166C4⋊Q8#6711yescharacteristic
167C4⋊Q8#6711yescharacteristic
168#127#6711yescharacteristic
169#125—21no—
170#124—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#121—21no—
174#121—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×D4—21no—
178C4×D4—21no—
179#117—21no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#106—22no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189#103—22no—
190#103—64no—
191#103—64no—
192#102—22no—
193#102—64no—
194C22⋊C8—32no—
195#97—61no—
196C4×C8—61no—
197C4×C8—61no—
198#95—61no—
199C23×C3—41no—
200C23×C3#12711yescharacteristic
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×D4—42no—
206C3×D4—41no—
207C3×D4—41no—
208C3×D4—81no—
209C3×D4—81no—
210C3×D4—81no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—82no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#13512yes—
225C2×C3×C4#13712yes—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—82no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231Dic6—44no—
232Dic6—82no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236#41—42no—
237#41—42no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8—242no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—244no—
255SD16—82no—
256M16—62no—
257M16—62no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—62no—
261C2×C8—62no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42—41no—
280C42—122no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39211yescharacteristic
291C22×C3#49511yescharacteristic
292C22×C3#49711yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4#45212yes—
300C3×C4#45511yescharacteristic
301C3×C4#45611yescharacteristic
302C3×C4#46412yes—
303Dic3—82no—
304Dic3—82no—
305C23—41no—
306C23#94011yescharacteristic
307Q8—42no—
308Q8—42no—
309Q8—41no—
310Q8—41no—
311Q8—124no—
312Q8—242no—
313D4—42no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—242no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4#91512yes—
332C2×C4#91612yes—
333C2×C4#93711yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#93811yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#95311yescharacteristic
339C8—42no—
340C8—122no—
341C8—122no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—161no—
345C2×C3#200711yescharacteristic
346C2×C3#200811yescharacteristic
347C2×C3#201511yescharacteristic
348C22—21no—
349C22—21no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22#516711yescharacteristic
354C22#516811yescharacteristic
355C22#522311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—242no—
363C4—242no—
364C4#516612yes—
365C4#517312yes—
366C4#524211yescharacteristic
367C4#524611yescharacteristic
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—161no—
372C2#7188111yescharacteristic
373C2#7205811yescharacteristic
374C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3246C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3246D3
hypocenterC3#10198
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
2. #83370C2semidirect
4. #71841C2semidirect
6. #70459C2semidirect
7. #67706C2semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
C3×#32462
#833702
#832832
#718412
#718112
#704592
#677062
#101983