Finite Group Databasebeta

← #34026398

Subgroups — #34026398

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11924037C212yesmaximal, complemented
2#22626125C214yesmaximal, complemented
3#22626124C212yesmaximal, complemented
4#22625796C212yesmaximal
5#22625795C211yesmaximal, characteristic
6#22625681C212yesmaximal
7C4×#14556651C211yesmaximal, characteristic
8C2×#17989295C211yesmaximal, characteristic
9#17989638—22no—
10#17989638—24no—
11Q8×#10596833—22no—
12#17989590C2212yes—
13#17989590C2214yes—
14C2×#14556651C2212yes—
15C2×#14556651C2211yescharacteristic
16C2×#14556651C2211yescharacteristic
17C4×#11924037C2212yes—
18C4×#11924037C2214yes—
19#17989467C2212yes—
20#17989467C2214yes—
21#17989467C2214yes—
22#17989467C2212yes—
23#17989295—22no—
24#17989295—22no—
25#17989295—24no—
26C2×#14556590C2211yescharacteristic
27C2×#14556581C2212yes—
28C2×#14556571C412yes—
29C2×#14556568C412yes—
30C2×#14556567C411yescharacteristic
31C2×Q8×Dic73C2211yescharacteristic
32C2×C4×C22⋊D73C2211yescharacteristic
33C2×#14556551C2212yes—
34C22×C4⋊Dic73C411yescharacteristic
35C2×C4⋊C4×D73C411yescharacteristic
36C2×C22⋊Q8×C73C411yescharacteristic
37C2×#11924037C2314yes—
38C2×#11924037C2312yes—
39C2×#11924037C2312yes—
40#14556658—24no—
41#14556658—22no—
42C4⋊#10596833—24no—
43C4⋊#10596833—22no—
44#14556651—24no—
45#14556651—22no—
46#14556651Q814yescomplemented
47#14556646—24no—
48#14556646—22no—
49C4×#10596833—24no—
50C4×#10596833—22no—
51C22×Dic146C2×C412yes—
52C22×C22⋊D73C2311yescharacteristic
53C23×Dic73C2×C412yes—
54C22×C4×D73C2×C411yescharacteristic
55C22×C4×D73C2×C412yes—
56#14556590—22no—
57#14556590—22no—
58#14556581—24no—
59#14556581—24no—
60#14556571—24no—
61#14556571—24no—
62#14556568—24no—
63#14556568—24no—
64#14556567—22no—
65#14556567—22no—
66Q8×Dic73D412yes—
67Q8×Dic73D412yes—
68Q8×Dic73D414yes—
69C4×C22⋊D73—22no—
70C4×C22⋊D73—22no—
71#14556551—24no—
72#14556551—24no—
73C2×C4⋊Dic73—22no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2×C411yescharacteristic
77C2×C4⋊Dic73C2×C412yes—
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×C4⋊Dic73C2×C412yes—
80C2×#11924007C2×C412yes—
81C2×#11924007C2×C412yes—
82C2×#11924007C2312yes—
83C2×#11924007C2311yescharacteristic
84C4⋊C4×D73—22no—
85C4⋊C4×D73—21no—
86C4⋊C4×D73—21no—
87C2×C22⋊Dic73C2×C411yescharacteristic
88C2×#11924005C2×C412yes—
89C2×#11924005C2×C411yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2312yes—
92C22×Q8×C73C2×C411yescharacteristic
93C23×C4×C73C2×C411yescharacteristic
94C22⋊Q8×C73—22no—
95C22⋊Q8×C73—22no—
96C2×C4⋊C4×C73C2×C411yescharacteristic
97C2×C4⋊C4×C73C2×C412yes—
98C2×C22⋊C4×C73C2×C412yes—
99#11924037—22no—
100#11924037—24no—
101#11924037—22no—
102#11924037—24no—
103#11924037—22no—
104#11924037C2×Q814yes—
105C2×#10596833—22no—
106C2×#10596833—24no—
107C2×#10596833—22no—
108C23×D73C22×C411yescharacteristic
109C2×Dic146—24no—
110C2×Dic146—82no—
111C2×C22⋊D73—22no—
112C2×C22⋊D73—22no—
113C2×C22⋊D73C2×Q812yes—
114C2×C22⋊D73#4112yes—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—82no—
118C22×Dic73C22×C412yes—
119C22×Dic73C22×C411yescharacteristic
120C22×Dic73C22×C411yescharacteristic
121C22×Dic73C22×C412yes—
122C22×Dic73C2411yescharacteristic
123C2×C4×D73—21no—
124C2×C4×D73—21no—
125C2×C4×D73—22no—
126C2×C4×D73—22no—
127C2×C4×D73—42no—
128C2×C4×D73—82no—
129C4⋊Dic73—21no—
130C4⋊Dic73—21no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73C22⋊C412yes—
134C4⋊Dic73C22⋊C411yescharacteristic
135C4⋊Dic73C22⋊C411yescharacteristic
136C4⋊Dic73C22⋊C414yes—
137C4⋊Dic73C22⋊C414yes—
138#11924007—24no—
139#11924007—22no—
140#11924007—22no—
141#11924007—24no—
142#11924007—22no—
143C22⋊Dic73—22no—
144#11924005—24no—
145#11924005—22no—
146C4×Dic73C2×D414yes—
147C4×Dic73C2×D414yes—
148C4×Dic73C2×D412yes—
149C4×Dic73C2×D412yes—
150C24×C73C22×C411yescharacteristic
151C2×Q8×C73—42no—
152C2×Q8×C73C22⋊C412yes—
153C2×Q8×C73C22⋊C412yes—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—42no—
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C412yes—
159C22×C4×C73C22×C411yescharacteristic
160C22×C4×C73C22×C412yes—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—24no—
164C22⋊C4×C73—24no—
165C23×C73—21no—
166C23×C73—21no—
167C23×C73—41no—
168C23×C73—41no—
169C23×C73C4×Q812yes—
170C23×C73C23×C411yescharacteristic
171C22×D73—21no—
172C22×D73—21no—
173C22×D73—41no—
174C22×D73—41no—
175C22×D73C4×Q812yes—
176Q8×C73—24no—
177Q8×C73—42no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—21no—
181C2×C4×C73—42no—
182C2×C4×C73—42no—
183C2×C4×C73—42no—
184C2×C4×C73—41no—
185C2×C4×C73—42no—
186C2×C4×C73C2×C22⋊C414yes—
187C2×C4×C73C2×C22⋊C411yescharacteristic
188C2×C4×C73C2×C22⋊C411yescharacteristic
189C22⋊D73—24no—
190C22⋊D73—24no—
191C2×Dic73—22no—
192C2×Dic73—21no—
193C2×Dic73—42no—
194C2×Dic73—41no—
195C2×Dic73—41no—
196C2×Dic73—42no—
197C2×Dic73—82no—
198C2×Dic73—82no—
199C2×Dic73C2×C22⋊C412yes—
200C2×Dic73C2×C22⋊C412yes—
201C2×Dic73C2×C22⋊C412yes—
202C2×Dic73C22×D412yes—
203C2×Dic73C22×Q811yescharacteristic
204C2×Dic73C2×#4111yescharacteristic
205C2×Dic732−1+412yes—
206C4×D73—42no—
207C4×D73—42no—
208C4×D73—82no—
209C4×D73—82no—
210Dic146—84no—
211#10596833—24no—
212#10596833—44no—
213#10596833—42no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—22no—
217C22×C73—42no—
218C22×C73—42no—
219C22×C73—41no—
220C22×C73—41no—
221C22×C73C22×C22⋊C411yescharacteristic
222C22×C73C2×C4×Q811yescharacteristic
223C22×C73#51911yescharacteristic
224C2×D73—22no—
225C2×D73—42no—
226C2×D73—41no—
227C2×D73—41no—
228C2×D73—42no—
229C4×C73—21no—
230C4×C73—24no—
231C4×C73—41no—
232C4×C73—42no—
233C4×C73#41312yes—
234Dic73—22no—
235Dic73—22no—
236Dic73—22no—
237Dic73—41no—
238Dic73—42no—
239Dic73#54812yes—
240Dic73D4×Q812yes—
241#22515—731nomaximal, Sylow
242C2×C73—21no—
243C2×C73—21no—
244C2×C73—42no—
245C2×C73Q8×C22⋊C411yescharacteristic
246C2×C73C2×#41311yescharacteristic
247C2×C73#288811yescharacteristic
248D73—42no—
249C2×#554—731no—
250Q8×M16—732no—
251#1998—734no—
252#1997—732no—
253#1800—731no—
254#1800—732no—
255C4×#100—731no—
256C4×#100—732no—
257C73#2251511yescharacteristic, complemented, Sylow
258#554—1462no—
259#554—1464no—
260#554—1462no—
261C2×C4⋊Q8—731no—
262C2×#125—732no—
263C2×C22⋊Q8—731no—
264C2×C22⋊Q8—732no—
265C2×C22⋊Q8—732no—
266C2×C22⋊Q8#6440811yescharacteristic
267C2×C4×Q8—731no—
268C2×C4×D4—731no—
269C2×C4×D4—732no—
270C22×C4⋊C4—731no—
271C22×C4⋊C4—731no—
272#446—1464no—
273#446—1462no—
274C8×Q8—1462no—
275#423—732no—
276#423—734no—
277C2×#100—731no—
278C2×#100—732no—
279C2×#100—731no—
280C4×M16—732no—
281C4×M16—734no—
282#343—734no—
283#343—732no—
284#343—734no—
285#343—732no—
286C22×Q8—732no—
287C22×Q8C2×#6440811yescharacteristic
288C22×D4—731no—
289C23×C4—731no—
290C23×C4—732no—
291C23×C4—732no—
292C23×C4C2×#6440811yescharacteristic
293C2×M16—732no—
294C2×M16—732no—
295C2×M16—734no—
296C4⋊Q8—1462no—
297C4⋊Q8—1462no—
298#125—1464no—
299#125—1464no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—1464no—
303C22⋊Q8—1464no—
304C22⋊Q8—1464no—
305C22⋊Q8—1464no—
306C22⋊Q8—1462no—
307C22⋊Q8—1462no—
308C4×Q8—732no—
309C4×Q8—734no—
310C4×Q8—732no—
311C4×D4—1464no—
312C4×D4—1464no—
313C4×D4—1462no—
314C4×D4—1462no—
315C2×C4⋊C4—731no—
316C2×C4⋊C4—732no—
317C2×C4⋊C4—732no—
318C2×C4⋊C4—732no—
319C2×C4⋊C4—731no—
320C2×C4⋊C4—731no—
321C2×C4⋊C4—732no—
322C2×C4⋊C4—732no—
323C2×C4⋊C4—1462no—
324C2×C4⋊C4—1461no—
325C2×C4⋊C4—1461no—
326C2×C4⋊C4—1462no—
327C2×C4⋊C4—1461no—
328C2×C4⋊C4—1461no—
329C2×C4⋊C4C2×#6440811yescharacteristic
330C2×C4⋊C4C2×#6440812yes—
331C2×C22⋊C4—731no—
332C2×C22⋊C4—731no—
333C2×C22⋊C4—732no—
334C2×C22⋊C4C2×#6440812yes—
335C2×C42—732no—
336C2×C42—731no—
337C4⋊C8—1464no—
338C4⋊C8—1462no—
339C4⋊C8—1464no—
340C4⋊C8—1462no—
341#100—734no—
342#100—1464no—
343#100—1462no—
344#97—1464no—
345#97—1462no—
346C4×C8—1464no—
347C4×C8—1462no—
348C24—731no—
349C24C22×#6440811yescharacteristic
350C2×Q8—42no—
351C2×Q8—5842no—
352C2×Q8—1464no—
353C2×Q8C22⋊#6440812yes—
354C2×Q8C22⋊#6440812yes—
355C2×D4—732no—
356C2×D4—732no—
357C2×D4—1462no—
358C2×D4—1462no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—42no—
362C22×C4—2922no—
363C22×C4—5842no—
364C22×C4—5842no—
365C22×C4—731no—
366C22×C4—732no—
367C22×C4—731no—
368C22×C4—732no—
369C22×C4—731no—
370C22×C4—1462no—
371C22×C4—1462no—
372C22×C4—1461no—
373C22×C4—1461no—
374C22×C4—1462no—
375C22×C4—1462no—
376C22×C4C22×#6440811yescharacteristic
377C22×C4C22×#6440811yescharacteristic
378C22×C4C22×#6440812yes—
379C22×C4C22×#6440812yes—
380M16—734no—
381M16—1462no—
382M16—1464no—
383M16—1462no—
384M16—1464no—
385M16—1462no—
386C2×C8—1464no—
387C2×C8—1462no—
388C2×C8—1462no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—24no—
392C4⋊C4—732no—
393C4⋊C4—731no—
394C4⋊C4—731no—
395C4⋊C4—734no—
396C4⋊C4—734no—
397C4⋊C4—1462no—
398C4⋊C4—1462no—
399C4⋊C4—1462no—
400C4⋊C4—1464no—
401C4⋊C4—1462no—
402C4⋊C4—1462no—
403C4⋊C4—1464no—
404C4⋊C4—1461no—
405C4⋊C4—1461no—
406C22⋊C4—24no—
407C22⋊C4—1462no—
408C22⋊C4—1462no—
409C22⋊C4—1464no—
410C42—734no—
411C42—732no—
412C42—734no—
413C42—732no—
414C23—21no—
415C23—21no—
416C23—41no—
417C23—41no—
418C23—2921no—
419C23—2921no—
420C23—732no—
421C23—1461no—
422C23—1461no—
423C23Q8×#6440812yes—
424C23C23×#6440811yescharacteristic
425Q8—24no—
426Q8—42no—
427Q8—5844no—
428D4—1464no—
429D4—1464no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—2922no—
439C2×C4—2922no—
440C2×C4—2921no—
441C2×C4—2922no—
442C2×C4—2922no—
443C2×C4—2921no—
444C2×C4—5842no—
445C2×C4—5842no—
446C2×C4—5842no—
447C2×C4—5842no—
448C2×C4—732no—
449C2×C4—731no—
450C2×C4—731no—
451C2×C4—732no—
452C2×C4—732no—
453C2×C4—732no—
454C2×C4—732no—
455C2×C4—1461no—
456C2×C4—1462no—
457C2×C4C2×C22⋊#6440814yes—
458C2×C4C2×C22⋊#6440811yescharacteristic
459C2×C4C2×C22⋊#6440811yescharacteristic
460C8—2924no—
461C8—2922no—
462C8—1464no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—2922no—
471C22—2922no—
472C22—2921no—
473C22—2921no—
474C22—1462no—
475C22C22×C22⋊#6440811yescharacteristic
476C22C2×Q8×#6440811yescharacteristic
477C22#1798965511yescharacteristic
478C4—21no—
479C4—24no—
480C4—41no—
481C4—42no—
482C4—2921no—
483C4—2922no—
484C4—732no—
485C4—732no—
486C4—1462no—
487C4—1462no—
488C4—1462no—
489C4#1798957412yes—
490C2—21no—
491C2—21no—
492C2—42no—
493C2—2922no—
494C2Q8×C22⋊#6440811yescharacteristic
495C2C2×#1798957411yescharacteristic
496C2#2262654211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×C22⋊D73C23
cosocleC22×Dic73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#22515
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#11924037C2semidirect
2. #22626125C2semidirect
3. #22626124C2semidirect
46. #14556651Q8semidirect
257. C73#22515semidirect

Maximal subgroups

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

subgroupindex
Q8×#119240372
#226261252
#226261242
#226257962
#226257952
#226256812
C4×#145566512
C2×#179892952
#2251573