Finite Group Databasebeta

← #10824698

Subgroups — #10824698

Order 768.

820 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#433C211yesmaximal, characteristic
2#82235C212yesmaximal
3C2×C4×Dic12C211yesmaximal, characteristic, complemented
4#70489C212yesmaximal, complemented
5#70487C211yesmaximal, characteristic, complemented
6#70462C211yesmaximal, characteristic, complemented
7#69964C211yesmaximal, characteristic
8#69938C211yesmaximal, characteristic
9#69937C212yesmaximal
10#69813C212yesmaximal, complemented
11C4×#818C211yesmaximal, characteristic, complemented
12#23705—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#924C2212yes—
15C22×Dic12C2211yescharacteristic
16C2×C4×Dic6C2212yes—
17C3×#433—24no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24#5212C2212yes—
25C2×#818C2212yes—
26C2×#818C2211yescharacteristic
27C2×#816C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#4978—22no—
31#4976—22no—
32#4945—22no—
33#4944—22no—
34C4×C4⋊Dic3C2212yes—
35#4929C2212yes—
36#4928C2212yes—
37C4×Dic12—24no—
38#4551C2212yes—
39#4551C2211yescharacteristic
40#4551C2212yes—
41#4551C2211yescharacteristic
42C2×C4×Q16—31no—
43C2×#433D311yescharacteristic
44#2323—32no—
45#2086—31no—
46#2075—31no—
47#2068—32no—
48#2056—31no—
49#2036—32no—
50#2001—31no—
51C4×#107—31no—
52#1763—32no—
53C23×C3×C4C2311yescharacteristic
54C22×Dic6C2312yes—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×#117—22no—
60C3×#117—22no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4D411yescharacteristic
64C2×C3×C22⋊C4D411yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×#217—22no—
68C2×#217—22no—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72#924—24no—
73#924—24no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×Dic12—24no—
80C2×Dic12—22no—
81C4×Dic6—24no—
82C4×Dic6—24no—
83C3×C4⋊C8—22no—
84C3×C4⋊C8—22no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8—22no—
87C3×#97—22no—
88C3×C4×C8D414yes—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94#818—22no—
95#818—24no—
96#816—24no—
97#816—24no—
98#816—24no—
99C22×Q16—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C2×#117C2×D311yescharacteristic
103C4×Q16—64no—
104#433—24no—
105C2×#107—32no—
106C2×#107—31no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×C4⋊C8C2×D311yescharacteristic
109C2×#103—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×C22⋊C8C2×D311yescharacteristic
113C2×C22⋊C8C2×D311yescharacteristic
114C2×#97C2×D311yescharacteristic
115C2×C4×C8C2×D311yescharacteristic
116#391—32no—
117#389—32no—
118#389—62no—
119#388—62no—
120#388—62no—
121#387—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340—31no—
127#340—32no—
128C24×C3C2×D411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6#4114yes—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8C2×D412yes—
156C2×C3×C8C2×D412yes—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C42—21no—
162C3×C42—21no—
163C3×C42C2×D412yes—
164C3×C42#4112yes—
165#217—28no—
166#217—82no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3#4114yes—
173Dic3⋊C4—28no—
174Dic3⋊C4—44no—
175Dic3⋊C4—82no—
176C4×Dic3—44no—
177C4×Dic3—82no—
178Dic12—28no—
179C22×Q8—32no—
180C23×C4C22×D311yescharacteristic
181C2×Q16—64no—
182C2×Q16—62no—
183C22×C8C22×D311yescharacteristic
184C22×C8C22×D311yescharacteristic
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187C22⋊Q8—64no—
188C22⋊Q8—64no—
189C4×Q8—64no—
190C4×Q8—64no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4D1211yescharacteristic
200C2×C4⋊C4D1211yescharacteristic
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4D1211yescharacteristic
204C2×C22⋊C4D1211yescharacteristic
205C2×C42—62no—
206C2×C42—62no—
207C2×C42C22×D311yescharacteristic
208C2×C42C22×D311yescharacteristic
209#107—64no—
210#107—62no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213#103—64no—
214#103—64no—
215#103—64no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#97—22no—
219C4×C8#6714yes—
220#95—62no—
221#95—62no—
222#95—62no—
223#95—62no—
224#95—62no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—22no—
229C23×C3—21no—
230C23×C3C22×D411yescharacteristic
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4#12112yes—
245C2×C3×C4#12112yes—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4112yes—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262Dic6—24no—
263Dic6—24no—
264Dic6—42no—
265Dic6—42no—
266C3×C8—22no—
267C3×C8—22no—
268C3×C8—41no—
269C3×C8—41no—
270C24C2×D1211yescharacteristic
271C2×Q8—34no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×Q8—62no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4C2×D1211yescharacteristic
286C22×C4C2×D1211yescharacteristic
287C22×C4C2×D1211yescharacteristic
288C22×C4C2×D1211yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4C23×D311yescharacteristic
291Q16—68no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8C2×#6712yes—
299C2×C8C2×#6712yes—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—34no—
303C4⋊C4—68no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—124no—
310C4⋊C4—242no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—68no—
314C22⋊C4—242no—
315C42—21no—
316C42—21no—
317C42—124no—
318C42—242no—
319C42#23512yes—
320C42C2×#6712yes—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C2×#12111yescharacteristic
327C22×C3#52512yes—
328C22×C3#57611yescharacteristic
329C22×C3#57812yes—
330C22×C3#57811yescharacteristic
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4#52511yescharacteristic
337C3×C4#52912yes—
338C3×C4#53211yescharacteristic
339Dic3—44no—
340Dic3—44no—
341Dic3—82no—
342Dic3—82no—
343C23—21no—
344C23—22no—
345C23—21no—
346C23—22no—
347C23—21no—
348C23C22×D1211yescharacteristic
349Q8—64no—
350Q8—64no—
351Q8—122no—
352Q8—122no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—64no—
367C2×C4—64no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4#93012yes—
379C2×C4#93012yes—
380C2×C4C2×#23511yescharacteristic
381C2×C4D3×#4112yes—
382C2×C4D3×#4112yes—
383C2×C4C22×#6711yescharacteristic
384C8—22no—
385C8—22no—
386C8—41no—
387C8—41no—
388C2×C3—81no—
389C2×C3#239211yescharacteristic
390C2×C3#305911yescharacteristic
391C2×C3#306111yescharacteristic
392C2×C3#312412yes—
393C2×C3#312712yes—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22#552412yes—
400C22#575211yescharacteristic
401C22#575312yes—
402C22#575311yescharacteristic
403C22C2×#93011yescharacteristic
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—124no—
410C4—124no—
411C4—242no—
412C4—242no—
413C4#552812yes—
414C4#582811yescharacteristic
415C4#583211yescharacteristic
416C3#2370511yescharacteristic, complemented, Sylow
417C2—81no—
418C2#7938812yes—
419C2#7939612yes—
420C2#8227311yescharacteristic
421C2#8380111yescharacteristic
422C2#8380711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#433C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#433D3
hypocenterC3#23705
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic12C2semidirect
4. #70489C2semidirect
5. #70487C2semidirect
6. #70462C2semidirect
10. #69813C2semidirect
11. C4×#818C2semidirect
416. C3#23705semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4332
#822352
C2×C4×Dic122
#704892
#704872
#704622
#699642
#699382
#699372
#698132
C4×#8182
#237053