Finite Group Databasebeta

← #10814902

Subgroups — #10814902

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79166C211yesmaximal, characteristic
2#78552C212yesmaximal, complemented
3C3×#1773C211yesmaximal, characteristic, complemented
4#70696C211yesmaximal, characteristic
5#70694C211yesmaximal, characteristic, complemented
6#70659C211yesmaximal, characteristic, complemented
7#69900C212yesmaximal, complemented
8#69845C211yesmaximal, characteristic
9#69811C211yesmaximal, characteristic, complemented
10#68770C212yesmaximal, complemented
11#68744C212yesmaximal, complemented
12#22804—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14#5522C2212yes—
15C2×C3×#108C2211yescharacteristic
16C3×#427C2211yescharacteristic
17C3×#405C2212yes—
18C2×C3×#97C2211yescharacteristic
19C2×#825C2212yes—
20#5129C2212yes—
21C2×#822C2212yes—
22#5118C2212yes—
23#5110C2212yes—
24C2×#819C2211yescharacteristic
25C2×#815C2211yescharacteristic
26C2×Dic3⋊C8C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5002C2212yes—
29#4997C2212yes—
30C4×Dic3⋊C4C2211yescharacteristic
31C2×#805C2212yes—
32#4889C412yescomplemented
33#4889C412yescomplemented
34#4869C412yescomplemented
35#4853C412yescomplemented
36#4709—22no—
37#4705—22no—
38#4690—22no—
39#4685—22no—
40C3×#339C2212yes—
41#4550C2211yescharacteristic
42#4532C2212yes—
43#4524C2211yescharacteristic
44#2943—31no—
45#2925—32no—
46#1975—31no—
47#1974—31no—
48#1929—32no—
49#1929—32no—
50#1849—31no—
51#1806—32no—
52#1773D311yescharacteristic, complemented
53#1772—31no—
54#1771—31no—
55C2×#235C2312yes—
56C2×C3×M16C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×#117C2312yes—
60C2×C3×C42C2311yescharacteristic
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3C2311yescharacteristic
63C2×C4×Dic3C2311yescharacteristic
64C2×#208C2312yes—
65C22×C3⋊C8C2311yescharacteristic
66C22×C3⋊C8C2311yescharacteristic
67C2×Dic12C2×C411yescharacteristic
68#904—22no—
69#904—22no—
70C2×D24C2×C411yescharacteristic
71C2×#204C2×C412yes—
72#893C2×C412yes—
73#893C2×C412yes—
74#887C2×C412yes—
75#887C2×C412yes—
76#67⋊C4—22no—
77#67⋊C4—22no—
78#872—22no—
79C42×D3—22no—
80C3×#108—22no—
81C3×#107C2×C411yescharacteristic
82C3×#107C2×C411yescharacteristic
83C3×#106C2×C412yes—
84C3×#97—22no—
85C3×#97—22no—
86C3×C4×C8—22no—
87C3⋊#95—21no—
88#825—42no—
89#822—42no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#819—22no—
95#815C2×C412yes—
96#815C2×C412yes—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99Dic3⋊C8D412yes—
100Dic3⋊C8D412yes—
101C8×Dic3—22no—
102#810C2×C412yes—
103#810C2×C412yes—
104#809C2×C412yes—
105#809C2×C412yes—
106#805—42no—
107#803—22no—
108C4×C3⋊C8—22no—
109C2×#135—31no—
110C4×#41—32no—
111#442—32no—
112#441—32no—
113#440—32no—
114#440—32no—
115#433—62no—
116#433—62no—
117#432—62no—
118#431—62no—
119C2×#108—31no—
120C2×#108—32no—
121C2×#108C2×D311yescharacteristic
122#427C2×D311yescharacteristic
123#423—32no—
124C2×C4⋊C8—31no—
125C2×#104—32no—
126C2×#104—32no—
127#416—32no—
128#408—32no—
129#408—32no—
130#405—32no—
131#405C2×D312yes—
132C2×#97—31no—
133C2×#97C2×D311yescharacteristic
134C2×C4×C8—31no—
135C4×C4⋊C4—31no—
136#339—32no—
137#339C2×D312yes—
138#336—31no—
139#336—31no—
140C22×C3×C4—21no—
141C22×C3×C4C2411yescharacteristic
142C2×#67—22no—
143C22×Dic3—21no—
144C22×Dic3#4111yescharacteristic
145C22×Dic3#4111yescharacteristic
146#235—22no—
147#235—22no—
148#235—22no—
149#235—22no—
150C2×D12C22×C412yes—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×Dic6C22×C412yes—
155C3×M16—22no—
156C3×M16—22no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—22no—
160C2×C3×C8C22×C411yescharacteristic
161C2×C3×C8C22×C411yescharacteristic
162C2×C3×C8C2×D412yes—
163C2×C3×C8#4112yes—
164C2×C3×C8#4112yes—
165C3×C4⋊C4C22×C412yes—
166C3×C4⋊C4C22×C412yes—
167C3×C22⋊C4—22no—
168C3×C42—22no—
169C3×C42—21no—
170C3×C42—21no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173Dic3⋊C4—22no—
174Dic3⋊C4—22no—
175Dic3⋊C4—42no—
176C4×Dic3—22no—
177C4×Dic3—22no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C4×Dic3C22×C412yes—
181C4×Dic3C2×D412yes—
182#208—22no—
183#208—22no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8C22×C412yes—
190C2×C3⋊C8C2×D411yescharacteristic
191C2×C3⋊C8C2×D411yescharacteristic
192Dic12—22no—
193D24—22no—
194#204—24no—
195C2×#41—32no—
196#135—62no—
197#135—62no—
198C2×Q16—31no—
199C2×SD16—32no—
200C2×D8—31no—
201C2×M16—32no—
202C2×M16C22×D312yes—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8C22×D311yescharacteristic
206C22×C8C22×D311yescharacteristic
207C4×Q8—32no—
208C4×Q8—32no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—62no—
212C4×D4—62no—
213#117—62no—
214#117C22×D312yes—
215C2×C4⋊C4—61no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—62no—
219C2×C42C22×D311yescharacteristic
220#108—22no—
221#108—62no—
222#108—122no—
223#107C4×D311yescharacteristic
224#107C4×D311yescharacteristic
225#106C4×D312yes—
226C4⋊C8—32no—
227C4⋊C8—32no—
228C4⋊C8—62no—
229C4⋊C8—62no—
230#104—122no—
231#104—122no—
232#103—32no—
233#103—32no—
234#102—32no—
235#102—32no—
236C22⋊C8—62no—
237C22⋊C8—62no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—22no—
241#97—22no—
242#97—32no—
243#97—32no—
244#97—62no—
245C4×C8—22no—
246C4×C8—62no—
247C4×C8—62no—
248#95—61no—
249C23×C3C2×#4111yescharacteristic
250C22×D3—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4C23×C411yescharacteristic
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4H(5,𝔽2)11yescharacteristic
261C2×C3×C42−1+411yescharacteristic
262#67—42no—
263#67—42no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3C4×D412yes—
271C2×Dic3C4×D412yes—
272D12—22no—
273D12—22no—
274C4×D3—44no—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—82no—
278Dic6—22no—
279Dic6—22no—
280C3×C8—22no—
281C3×C8—42no—
282C3×C8#11712yes—
283C3×C8#11712yes—
284C3⋊C8—22no—
285C3⋊C8—21no—
286C3⋊C8—21no—
287C3⋊C8—42no—
288#41—62no—
289#41—62no—
290#41—62no—
291#41—62no—
292C2×Q8—32no—
293C2×D4—32no—
294C2×D4—62no—
295C22×C4—21no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4C23×D311yescharacteristic
303Q16—62no—
304SD16—64no—
305D8—62no—
306M16—22no—
307M16—22no—
308M16—62no—
309M16—62no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—32no—
314C2×C8—31no—
315C2×C8—31no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—62no—
321C2×C8C2×C4×D311yescharacteristic
322C2×C8C2×C4×D311yescharacteristic
323C2×C8#23512yes—
324C2×C8#23512yes—
325C2×C8D3×D412yes—
326C4⋊C4—62no—
327C4⋊C4—62no—
328C4⋊C4—122no—
329C4⋊C4C2×C4×D312yes—
330C4⋊C4C2×C4×D312yes—
331C22⋊C4—22no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C42—22no—
335C42—21no—
336C42—21no—
337C42—32no—
338C42—32no—
339C42—62no—
340C42—62no—
341C42—61no—
342C42—61no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3C2×C4×D411yescharacteristic
346C22×C3#55511yescharacteristic
347C22×C3#55611yescharacteristic
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—81no—
354C3×C4C2×#11711yescharacteristic
355C3×C4#52011yescharacteristic
356C3×C4#54711yescharacteristic
357C3×C4#55011yescharacteristic
358Dic3—22no—
359Dic3—22no—
360Dic3—42no—
361Dic3—42no—
362C23—62no—
363C23D3×#4111yescharacteristic
364Q8—62no—
365Q8—62no—
366D4—62no—
367D4—62no—
368D4—122no—
369D4—122no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—124no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—242no—
387C2×C4C22×C4×D311yescharacteristic
388C2×C4C2×#23511yescharacteristic
389C2×C4C2×#23511yescharacteristic
390C2×C4C2×D3×D411yescharacteristic
391C2×C4#100711yescharacteristic
392C2×C4#100911yescharacteristic
393C8—22no—
394C8—42no—
395C8—61no—
396C8—61no—
397C8—62no—
398C8—122no—
399C8#87212yes—
400C8#87212yes—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#235411yescharacteristic
404C2×C3#295511yescharacteristic
405C2×C3#295611yescharacteristic
406D3—82no—
407C22—21no—
408C22—21no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22#553211yescharacteristic
413C22C4×D3×D411yescharacteristic
414C22#556811yescharacteristic
415C4—42no—
416C4—42no—
417C4—62no—
418C4—62no—
419C4—81no—
420C4—122no—
421C4—122no—
422C4C2×#87211yescharacteristic
423C4#550711yescharacteristic
424C4#555311yescharacteristic
425C4#557411yescharacteristic
426C3#2280411yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—242no—
430C2#7363911yescharacteristic
431C2#7989711yescharacteristic
432C2#7989811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#22804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #78552C2semidirect
3. C3×#1773C2semidirect
5. #70694C2semidirect
6. #70659C2semidirect
7. #69900C2semidirect
9. #69811C2semidirect
10. #68770C2semidirect
11. #68744C2semidirect
32. #4889C4semidirect
33. #4889C4semidirect
34. #4869C4semidirect
35. #4853C4semidirect
52. #1773D3semidirect
426. C3#22804semidirect

Maximal subgroups

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

subgroupindex
#791662
#785522
C3×#17732
#706962
#706942
#706592
#699002
#698452
#698112
#687702
#687442
#228043