Finite Group Databasebeta

← #10697187

Subgroups — #10697187

Order 768.

508 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3077C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic, complemented
3C2×#5008C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic
6#70508C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#10182—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#478—22no—
13C3×#476—21no—
14C3×#474—21no—
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#418—21no—
18C2×#832—21no—
19#5093—21no—
20#5092—21no—
21#5050—22no—
22#5031—22no—
23#5008—22no—
24#5008—22no—
25C2×#808—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—22no—
30#4525—22no—
31#4486—22no—
32#3209—31no—
33#3077D311yescharacteristic
34C2×#476—31no—
35#1549—31no—
36#1481—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C22×C3⋊C8—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C3×#107—22no—
67C3×#106—22no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×#102—22no—
71C3×#102—22no—
72#832—24no—
73#830—21no—
74#830D412yes—
75#810—24no—
76#808—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×C4×D4C2×D311yescharacteristic
86#494—61no—
87#487—61no—
88#478—22no—
89#476—21no—
90#476—62no—
91#476—62no—
92#474—21no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431—31no—
97#428—21no—
98C2×#106—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101#418—21no—
102C2×#102—61no—
103#332—62no—
104Q8⋊C8—62no—
105#323—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4Q1612yescomplemented
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—82no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—41no—
123C2×Dic6—22no—
124C2×Dic6—41no—
125C2×Dic6—82no—
126C3×M16—41no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#216—84no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C22×Q8—61no—
156C22×D4#6711yescharacteristic
157C23×C4—21no—
158C2×Q16—64no—
159C2×M16—21no—
160C22×C8—61no—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8#6711yescharacteristic
166C4⋊Q8#6711yescharacteristic
167#125—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176#117—21no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C42C22×D311yescharacteristic
182#107—22no—
183#106—22no—
184#106—64no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—64no—
192#102—22no—
193#102—22no—
194#102—64no—
195C22⋊C8—32no—
196C22⋊C8—61no—
197C4×C8—61no—
198C4×C8—61no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3#12111yescharacteristic
205C3×Q8—42no—
206C3×Q8—42no—
207C3×D4—22no—
208C3×D4—22no—
209C3×D4—41no—
210C3×D4—41no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4—81no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×C3×C4C2×Q1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—42no—
230C2×Dic3—82no—
231Dic6—44no—
232Dic6—82no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8—242no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4#21612yescomplemented
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—121no—
254C22×C4D3×D411yescharacteristic
255C22×C4#23711yescharacteristic
256C22×C4C2×#6711yescharacteristic
257Q16—244no—
258M16—41no—
259M16—41no—
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—41no—
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42C2×#6711yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—42no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#46711yescharacteristic
295C22×C3#47011yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4#45111yescharacteristic
303C3×C4#45111yescharacteristic
304C3×C4#45211yescharacteristic
305C3×C4#45211yescharacteristic
306C3×C4#47112yes—
307Dic3—82no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—81no—
312C23—81no—
313C23#93911yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—124no—
317Q8—242no—
318D4—22no—
319D4—22no—
320D4—41no—
321D4—41no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4#93711yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#93811yescharacteristic
335C2×C4#93911yescharacteristic
336C2×C4#94011yescharacteristic
337C2×C4#94211yescharacteristic
338C2×C4C2×#21611yescharacteristic
339C2×C4#95011yescharacteristic
340C2×C4#95111yescharacteristic
341C2×C4#95311yescharacteristic
342C8—42no—
343C8—122no—
344C8—122no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—82no—
349C2×C3#200111yescharacteristic
350C2×C3#200811yescharacteristic
351C2×C3#201911yescharacteristic
352C22—21no—
353C22—21no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22#505111yescharacteristic
360C22#505211yescharacteristic
361C22#522311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—242no—
369C4#504711yescharacteristic
370C4#505011yescharacteristic
371C4#520012yes—
372C4#522811yescharacteristic
373C4#523911yescharacteristic
374C3#1018211yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—82no—
378C2#7170011yescharacteristic
379C2#7209711yescharacteristic
380C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3077D3
hypocenterC3#10182
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #83370C2semidirect
4. #71824C2semidirect
6. #70508C2semidirect
7. #67714C2semidirect
114. C2×C3×D4Q16semidirect
248. C2×D4#216semidirect
374. C3#10182semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833702
C2×#50082
#718242
#716012
#705082
#677142
#101823